Properties

Label 320.4.a.b
Level $320$
Weight $4$
Character orbit 320.a
Self dual yes
Analytic conductor $18.881$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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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: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 10)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - 8 q^{3} - 5 q^{5} + 4 q^{7} + 37 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 8 q^{3} - 5 q^{5} + 4 q^{7} + 37 q^{9} + 12 q^{11} + 58 q^{13} + 40 q^{15} + 66 q^{17} - 100 q^{19} - 32 q^{21} - 132 q^{23} + 25 q^{25} - 80 q^{27} + 90 q^{29} - 152 q^{31} - 96 q^{33} - 20 q^{35} + 34 q^{37} - 464 q^{39} - 438 q^{41} + 32 q^{43} - 185 q^{45} + 204 q^{47} - 327 q^{49} - 528 q^{51} - 222 q^{53} - 60 q^{55} + 800 q^{57} + 420 q^{59} - 902 q^{61} + 148 q^{63} - 290 q^{65} - 1024 q^{67} + 1056 q^{69} - 432 q^{71} + 362 q^{73} - 200 q^{75} + 48 q^{77} + 160 q^{79} - 359 q^{81} + 72 q^{83} - 330 q^{85} - 720 q^{87} + 810 q^{89} + 232 q^{91} + 1216 q^{93} + 500 q^{95} + 1106 q^{97} + 444 q^{99}+O(q^{100}) \) 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
0
0 −8.00000 0 −5.00000 0 4.00000 0 37.0000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 320.4.a.b 1
4.b odd 2 1 320.4.a.m 1
5.b even 2 1 1600.4.a.bx 1
8.b even 2 1 80.4.a.f 1
8.d odd 2 1 10.4.a.a 1
16.e even 4 2 1280.4.d.g 2
16.f odd 4 2 1280.4.d.j 2
20.d odd 2 1 1600.4.a.d 1
24.f even 2 1 90.4.a.a 1
24.h odd 2 1 720.4.a.j 1
40.e odd 2 1 50.4.a.c 1
40.f even 2 1 400.4.a.b 1
40.i odd 4 2 400.4.c.c 2
40.k even 4 2 50.4.b.a 2
56.e even 2 1 490.4.a.o 1
56.k odd 6 2 490.4.e.i 2
56.m even 6 2 490.4.e.a 2
72.l even 6 2 810.4.e.w 2
72.p odd 6 2 810.4.e.c 2
88.g even 2 1 1210.4.a.b 1
104.h odd 2 1 1690.4.a.a 1
120.m even 2 1 450.4.a.q 1
120.q odd 4 2 450.4.c.d 2
280.n even 2 1 2450.4.a.b 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
10.4.a.a 1 8.d odd 2 1
50.4.a.c 1 40.e odd 2 1
50.4.b.a 2 40.k even 4 2
80.4.a.f 1 8.b even 2 1
90.4.a.a 1 24.f even 2 1
320.4.a.b 1 1.a even 1 1 trivial
320.4.a.m 1 4.b odd 2 1
400.4.a.b 1 40.f even 2 1
400.4.c.c 2 40.i odd 4 2
450.4.a.q 1 120.m even 2 1
450.4.c.d 2 120.q odd 4 2
490.4.a.o 1 56.e even 2 1
490.4.e.a 2 56.m even 6 2
490.4.e.i 2 56.k odd 6 2
720.4.a.j 1 24.h odd 2 1
810.4.e.c 2 72.p odd 6 2
810.4.e.w 2 72.l even 6 2
1210.4.a.b 1 88.g even 2 1
1280.4.d.g 2 16.e even 4 2
1280.4.d.j 2 16.f odd 4 2
1600.4.a.d 1 20.d odd 2 1
1600.4.a.bx 1 5.b even 2 1
1690.4.a.a 1 104.h odd 2 1
2450.4.a.b 1 280.n even 2 1

Hecke kernels

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

\( T_{3} + 8 \) Copy content Toggle raw display
\( T_{7} - 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \) Copy content Toggle raw display
$3$ \( T + 8 \) Copy content Toggle raw display
$5$ \( T + 5 \) Copy content Toggle raw display
$7$ \( T - 4 \) Copy content Toggle raw display
$11$ \( T - 12 \) Copy content Toggle raw display
$13$ \( T - 58 \) Copy content Toggle raw display
$17$ \( T - 66 \) Copy content Toggle raw display
$19$ \( T + 100 \) Copy content Toggle raw display
$23$ \( T + 132 \) Copy content Toggle raw display
$29$ \( T - 90 \) Copy content Toggle raw display
$31$ \( T + 152 \) Copy content Toggle raw display
$37$ \( T - 34 \) Copy content Toggle raw display
$41$ \( T + 438 \) Copy content Toggle raw display
$43$ \( T - 32 \) Copy content Toggle raw display
$47$ \( T - 204 \) Copy content Toggle raw display
$53$ \( T + 222 \) Copy content Toggle raw display
$59$ \( T - 420 \) Copy content Toggle raw display
$61$ \( T + 902 \) Copy content Toggle raw display
$67$ \( T + 1024 \) Copy content Toggle raw display
$71$ \( T + 432 \) Copy content Toggle raw display
$73$ \( T - 362 \) Copy content Toggle raw display
$79$ \( T - 160 \) Copy content Toggle raw display
$83$ \( T - 72 \) Copy content Toggle raw display
$89$ \( T - 810 \) Copy content Toggle raw display
$97$ \( T - 1106 \) Copy content Toggle raw display
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