Properties

Label 320.4.a.r
Level $320$
Weight $4$
Character orbit 320.a
Self dual yes
Analytic conductor $18.881$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 320 = 2^{6} \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 320.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(18.8806112018\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{13}) \)
Defining polynomial: \( x^{2} - x - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 160)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 2\sqrt{13}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta q^{3} + 5 q^{5} + \beta q^{7} + 25 q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - \beta q^{3} + 5 q^{5} + \beta q^{7} + 25 q^{9} - 6 \beta q^{11} - 34 q^{13} - 5 \beta q^{15} + 114 q^{17} - 52 q^{21} - 29 \beta q^{23} + 25 q^{25} + 2 \beta q^{27} + 26 q^{29} + 14 \beta q^{31} + 312 q^{33} + 5 \beta q^{35} + 150 q^{37} + 34 \beta q^{39} + 342 q^{41} + 63 \beta q^{43} + 125 q^{45} + 81 \beta q^{47} - 291 q^{49} - 114 \beta q^{51} + 262 q^{53} - 30 \beta q^{55} + 68 \beta q^{59} + 262 q^{61} + 25 \beta q^{63} - 170 q^{65} + 69 \beta q^{67} + 1508 q^{69} - 146 \beta q^{71} + 682 q^{73} - 25 \beta q^{75} - 312 q^{77} - 28 \beta q^{79} - 779 q^{81} - 21 \beta q^{83} + 570 q^{85} - 26 \beta q^{87} - 630 q^{89} - 34 \beta q^{91} - 728 q^{93} - 966 q^{97} - 150 \beta q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 10 q^{5} + 50 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 10 q^{5} + 50 q^{9} - 68 q^{13} + 228 q^{17} - 104 q^{21} + 50 q^{25} + 52 q^{29} + 624 q^{33} + 300 q^{37} + 684 q^{41} + 250 q^{45} - 582 q^{49} + 524 q^{53} + 524 q^{61} - 340 q^{65} + 3016 q^{69} + 1364 q^{73} - 624 q^{77} - 1558 q^{81} + 1140 q^{85} - 1260 q^{89} - 1456 q^{93} - 1932 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
2.30278
−1.30278
0 −7.21110 0 5.00000 0 7.21110 0 25.0000 0
1.2 0 7.21110 0 5.00000 0 −7.21110 0 25.0000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(5\) \(-1\)

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 320.4.a.r 2
4.b odd 2 1 inner 320.4.a.r 2
5.b even 2 1 1600.4.a.ci 2
8.b even 2 1 160.4.a.e 2
8.d odd 2 1 160.4.a.e 2
16.e even 4 2 1280.4.d.t 4
16.f odd 4 2 1280.4.d.t 4
20.d odd 2 1 1600.4.a.ci 2
24.f even 2 1 1440.4.a.bd 2
24.h odd 2 1 1440.4.a.bd 2
40.e odd 2 1 800.4.a.q 2
40.f even 2 1 800.4.a.q 2
40.i odd 4 2 800.4.c.h 4
40.k even 4 2 800.4.c.h 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
160.4.a.e 2 8.b even 2 1
160.4.a.e 2 8.d odd 2 1
320.4.a.r 2 1.a even 1 1 trivial
320.4.a.r 2 4.b odd 2 1 inner
800.4.a.q 2 40.e odd 2 1
800.4.a.q 2 40.f even 2 1
800.4.c.h 4 40.i odd 4 2
800.4.c.h 4 40.k even 4 2
1280.4.d.t 4 16.e even 4 2
1280.4.d.t 4 16.f odd 4 2
1440.4.a.bd 2 24.f even 2 1
1440.4.a.bd 2 24.h odd 2 1
1600.4.a.ci 2 5.b even 2 1
1600.4.a.ci 2 20.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(320))\):

\( T_{3}^{2} - 52 \) Copy content Toggle raw display
\( T_{7}^{2} - 52 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} - 52 \) Copy content Toggle raw display
$5$ \( (T - 5)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} - 52 \) Copy content Toggle raw display
$11$ \( T^{2} - 1872 \) Copy content Toggle raw display
$13$ \( (T + 34)^{2} \) Copy content Toggle raw display
$17$ \( (T - 114)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} \) Copy content Toggle raw display
$23$ \( T^{2} - 43732 \) Copy content Toggle raw display
$29$ \( (T - 26)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} - 10192 \) Copy content Toggle raw display
$37$ \( (T - 150)^{2} \) Copy content Toggle raw display
$41$ \( (T - 342)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} - 206388 \) Copy content Toggle raw display
$47$ \( T^{2} - 341172 \) Copy content Toggle raw display
$53$ \( (T - 262)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} - 240448 \) Copy content Toggle raw display
$61$ \( (T - 262)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} - 247572 \) Copy content Toggle raw display
$71$ \( T^{2} - 1108432 \) Copy content Toggle raw display
$73$ \( (T - 682)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} - 40768 \) Copy content Toggle raw display
$83$ \( T^{2} - 22932 \) Copy content Toggle raw display
$89$ \( (T + 630)^{2} \) Copy content Toggle raw display
$97$ \( (T + 966)^{2} \) Copy content Toggle raw display
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