Properties

Label 320.6.a.k
Level $320$
Weight $6$
Character orbit 320.a
Self dual yes
Analytic conductor $51.323$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

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

Newform invariants

Self dual: yes
Analytic conductor: \(51.3228223402\)
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

\(f(q)\) \(=\) \( q + 6 q^{3} + 25 q^{5} + 118 q^{7} - 207 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 6 q^{3} + 25 q^{5} + 118 q^{7} - 207 q^{9} + 192 q^{11} - 1106 q^{13} + 150 q^{15} + 762 q^{17} - 2740 q^{19} + 708 q^{21} - 1566 q^{23} + 625 q^{25} - 2700 q^{27} - 5910 q^{29} + 6868 q^{31} + 1152 q^{33} + 2950 q^{35} + 5518 q^{37} - 6636 q^{39} - 378 q^{41} - 2434 q^{43} - 5175 q^{45} - 13122 q^{47} - 2883 q^{49} + 4572 q^{51} + 9174 q^{53} + 4800 q^{55} - 16440 q^{57} - 34980 q^{59} + 9838 q^{61} - 24426 q^{63} - 27650 q^{65} + 33722 q^{67} - 9396 q^{69} - 70212 q^{71} + 21986 q^{73} + 3750 q^{75} + 22656 q^{77} - 4520 q^{79} + 34101 q^{81} - 109074 q^{83} + 19050 q^{85} - 35460 q^{87} + 38490 q^{89} - 130508 q^{91} + 41208 q^{93} - 68500 q^{95} - 1918 q^{97} - 39744 q^{99}+O(q^{100}) \) 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 6.00000 0 25.0000 0 118.000 0 −207.000 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 320.6.a.k 1
4.b odd 2 1 320.6.a.f 1
8.b even 2 1 80.6.a.c 1
8.d odd 2 1 10.6.a.c 1
24.f even 2 1 90.6.a.b 1
24.h odd 2 1 720.6.a.v 1
40.e odd 2 1 50.6.a.b 1
40.f even 2 1 400.6.a.i 1
40.i odd 4 2 400.6.c.i 2
40.k even 4 2 50.6.b.b 2
56.e even 2 1 490.6.a.k 1
120.m even 2 1 450.6.a.u 1
120.q odd 4 2 450.6.c.f 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
10.6.a.c 1 8.d odd 2 1
50.6.a.b 1 40.e odd 2 1
50.6.b.b 2 40.k even 4 2
80.6.a.c 1 8.b even 2 1
90.6.a.b 1 24.f even 2 1
320.6.a.f 1 4.b odd 2 1
320.6.a.k 1 1.a even 1 1 trivial
400.6.a.i 1 40.f even 2 1
400.6.c.i 2 40.i odd 4 2
450.6.a.u 1 120.m even 2 1
450.6.c.f 2 120.q odd 4 2
490.6.a.k 1 56.e even 2 1
720.6.a.v 1 24.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} - 6 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(320))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \) Copy content Toggle raw display
$3$ \( T - 6 \) Copy content Toggle raw display
$5$ \( T - 25 \) Copy content Toggle raw display
$7$ \( T - 118 \) Copy content Toggle raw display
$11$ \( T - 192 \) Copy content Toggle raw display
$13$ \( T + 1106 \) Copy content Toggle raw display
$17$ \( T - 762 \) Copy content Toggle raw display
$19$ \( T + 2740 \) Copy content Toggle raw display
$23$ \( T + 1566 \) Copy content Toggle raw display
$29$ \( T + 5910 \) Copy content Toggle raw display
$31$ \( T - 6868 \) Copy content Toggle raw display
$37$ \( T - 5518 \) Copy content Toggle raw display
$41$ \( T + 378 \) Copy content Toggle raw display
$43$ \( T + 2434 \) Copy content Toggle raw display
$47$ \( T + 13122 \) Copy content Toggle raw display
$53$ \( T - 9174 \) Copy content Toggle raw display
$59$ \( T + 34980 \) Copy content Toggle raw display
$61$ \( T - 9838 \) Copy content Toggle raw display
$67$ \( T - 33722 \) Copy content Toggle raw display
$71$ \( T + 70212 \) Copy content Toggle raw display
$73$ \( T - 21986 \) Copy content Toggle raw display
$79$ \( T + 4520 \) Copy content Toggle raw display
$83$ \( T + 109074 \) Copy content Toggle raw display
$89$ \( T - 38490 \) Copy content Toggle raw display
$97$ \( T + 1918 \) Copy content Toggle raw display
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