Properties

Label 320.6.a.t
Level $320$
Weight $6$
Character orbit 320.a
Self dual yes
Analytic conductor $51.323$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $2$

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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: \(2\)
Coefficient field: \(\Q(\sqrt{5}) \)
Defining polynomial: \( x^{2} - x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
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 \(\beta = 2\sqrt{5}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 3 \beta q^{3} + 25 q^{5} + 31 \beta q^{7} - 63 q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - 3 \beta q^{3} + 25 q^{5} + 31 \beta q^{7} - 63 q^{9} - 58 \beta q^{11} - 154 q^{13} - 75 \beta q^{15} + 178 q^{17} - 216 \beta q^{19} - 1860 q^{21} + 589 \beta q^{23} + 625 q^{25} + 918 \beta q^{27} - 4110 q^{29} + 706 \beta q^{31} + 3480 q^{33} + 775 \beta q^{35} - 7442 q^{37} + 462 \beta q^{39} + 7270 q^{41} + 4005 \beta q^{43} - 1575 q^{45} - 1657 \beta q^{47} + 2413 q^{49} - 534 \beta q^{51} - 32226 q^{53} - 1450 \beta q^{55} + 12960 q^{57} - 7612 \beta q^{59} - 26770 q^{61} - 1953 \beta q^{63} - 3850 q^{65} - 11137 \beta q^{67} - 35340 q^{69} + 12098 \beta q^{71} - 18534 q^{73} - 1875 \beta q^{75} - 35960 q^{77} - 19396 \beta q^{79} - 39771 q^{81} + 17585 \beta q^{83} + 4450 q^{85} + 12330 \beta q^{87} - 107590 q^{89} - 4774 \beta q^{91} - 42360 q^{93} - 5400 \beta q^{95} - 108838 q^{97} + 3654 \beta q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 50 q^{5} - 126 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 50 q^{5} - 126 q^{9} - 308 q^{13} + 356 q^{17} - 3720 q^{21} + 1250 q^{25} - 8220 q^{29} + 6960 q^{33} - 14884 q^{37} + 14540 q^{41} - 3150 q^{45} + 4826 q^{49} - 64452 q^{53} + 25920 q^{57} - 53540 q^{61} - 7700 q^{65} - 70680 q^{69} - 37068 q^{73} - 71920 q^{77} - 79542 q^{81} + 8900 q^{85} - 215180 q^{89} - 84720 q^{93} - 217676 q^{97}+O(q^{100}) \) 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
1.61803
−0.618034
0 −13.4164 0 25.0000 0 138.636 0 −63.0000 0
1.2 0 13.4164 0 25.0000 0 −138.636 0 −63.0000 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 320.6.a.t 2
4.b odd 2 1 inner 320.6.a.t 2
8.b even 2 1 160.6.a.b 2
8.d odd 2 1 160.6.a.b 2
40.e odd 2 1 800.6.a.i 2
40.f even 2 1 800.6.a.i 2
40.i odd 4 2 800.6.c.h 4
40.k even 4 2 800.6.c.h 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
160.6.a.b 2 8.b even 2 1
160.6.a.b 2 8.d odd 2 1
320.6.a.t 2 1.a even 1 1 trivial
320.6.a.t 2 4.b odd 2 1 inner
800.6.a.i 2 40.e odd 2 1
800.6.a.i 2 40.f even 2 1
800.6.c.h 4 40.i odd 4 2
800.6.c.h 4 40.k even 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} - 180 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(320))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} - 180 \) Copy content Toggle raw display
$5$ \( (T - 25)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} - 19220 \) Copy content Toggle raw display
$11$ \( T^{2} - 67280 \) Copy content Toggle raw display
$13$ \( (T + 154)^{2} \) Copy content Toggle raw display
$17$ \( (T - 178)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} - 933120 \) Copy content Toggle raw display
$23$ \( T^{2} - 6938420 \) Copy content Toggle raw display
$29$ \( (T + 4110)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} - 9968720 \) Copy content Toggle raw display
$37$ \( (T + 7442)^{2} \) Copy content Toggle raw display
$41$ \( (T - 7270)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} - 320800500 \) Copy content Toggle raw display
$47$ \( T^{2} - 54912980 \) Copy content Toggle raw display
$53$ \( (T + 32226)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} - 1158850880 \) Copy content Toggle raw display
$61$ \( (T + 26770)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} - 2480655380 \) Copy content Toggle raw display
$71$ \( T^{2} - 2927232080 \) Copy content Toggle raw display
$73$ \( (T + 18534)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} - 7524096320 \) Copy content Toggle raw display
$83$ \( T^{2} - 6184644500 \) Copy content Toggle raw display
$89$ \( (T + 107590)^{2} \) Copy content Toggle raw display
$97$ \( (T + 108838)^{2} \) Copy content Toggle raw display
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