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: \(\Q(\sqrt{5}) \)
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 \(\beta = \frac{1}{2}(1 + \sqrt{5})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 6 - 12 \beta ) q^{3} + 25 q^{5} + ( -62 + 124 \beta ) q^{7} -63 q^{9} +O(q^{10})\) \( q + ( 6 - 12 \beta ) q^{3} + 25 q^{5} + ( -62 + 124 \beta ) q^{7} -63 q^{9} + ( 116 - 232 \beta ) q^{11} -154 q^{13} + ( 150 - 300 \beta ) q^{15} + 178 q^{17} + ( 432 - 864 \beta ) q^{19} -1860 q^{21} + ( -1178 + 2356 \beta ) q^{23} + 625 q^{25} + ( -1836 + 3672 \beta ) q^{27} -4110 q^{29} + ( -1412 + 2824 \beta ) q^{31} + 3480 q^{33} + ( -1550 + 3100 \beta ) q^{35} -7442 q^{37} + ( -924 + 1848 \beta ) q^{39} + 7270 q^{41} + ( -8010 + 16020 \beta ) q^{43} -1575 q^{45} + ( 3314 - 6628 \beta ) q^{47} + 2413 q^{49} + ( 1068 - 2136 \beta ) q^{51} -32226 q^{53} + ( 2900 - 5800 \beta ) q^{55} + 12960 q^{57} + ( 15224 - 30448 \beta ) q^{59} -26770 q^{61} + ( 3906 - 7812 \beta ) q^{63} -3850 q^{65} + ( 22274 - 44548 \beta ) q^{67} -35340 q^{69} + ( -24196 + 48392 \beta ) q^{71} -18534 q^{73} + ( 3750 - 7500 \beta ) q^{75} -35960 q^{77} + ( 38792 - 77584 \beta ) q^{79} -39771 q^{81} + ( -35170 + 70340 \beta ) q^{83} + 4450 q^{85} + ( -24660 + 49320 \beta ) q^{87} -107590 q^{89} + ( 9548 - 19096 \beta ) q^{91} -42360 q^{93} + ( 10800 - 21600 \beta ) q^{95} -108838 q^{97} + ( -7308 + 14616 \beta ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 50q^{5} - 126q^{9} + O(q^{10}) \) \( 2q + 50q^{5} - 126q^{9} - 308q^{13} + 356q^{17} - 3720q^{21} + 1250q^{25} - 8220q^{29} + 6960q^{33} - 14884q^{37} + 14540q^{41} - 3150q^{45} + 4826q^{49} - 64452q^{53} + 25920q^{57} - 53540q^{61} - 7700q^{65} - 70680q^{69} - 37068q^{73} - 71920q^{77} - 79542q^{81} + 8900q^{85} - 215180q^{89} - 84720q^{93} - 217676q^{97} + O(q^{100}) \)

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:

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

Atkin-Lehner signs

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

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 + 306 T^{2} + 59049 T^{4} \)
$5$ \( ( 1 - 25 T )^{2} \)
$7$ \( 1 + 14394 T^{2} + 282475249 T^{4} \)
$11$ \( 1 + 254822 T^{2} + 25937424601 T^{4} \)
$13$ \( ( 1 + 154 T + 371293 T^{2} )^{2} \)
$17$ \( ( 1 - 178 T + 1419857 T^{2} )^{2} \)
$19$ \( 1 + 4019078 T^{2} + 6131066257801 T^{4} \)
$23$ \( 1 + 5934266 T^{2} + 41426511213649 T^{4} \)
$29$ \( ( 1 + 4110 T + 20511149 T^{2} )^{2} \)
$31$ \( 1 + 47289582 T^{2} + 819628286980801 T^{4} \)
$37$ \( ( 1 + 7442 T + 69343957 T^{2} )^{2} \)
$41$ \( ( 1 - 7270 T + 115856201 T^{2} )^{2} \)
$43$ \( 1 - 26783614 T^{2} + 21611482313284249 T^{4} \)
$47$ \( 1 + 403777034 T^{2} + 52599132235830049 T^{4} \)
$53$ \( ( 1 + 32226 T + 418195493 T^{2} )^{2} \)
$59$ \( 1 + 270997718 T^{2} + 511116753300641401 T^{4} \)
$61$ \( ( 1 + 26770 T + 844596301 T^{2} )^{2} \)
$67$ \( 1 + 219594834 T^{2} + 1822837804551761449 T^{4} \)
$71$ \( 1 + 681226622 T^{2} + 3255243551009881201 T^{4} \)
$73$ \( ( 1 + 18534 T + 2073071593 T^{2} )^{2} \)
$79$ \( 1 - 1369983522 T^{2} + 9468276082626847201 T^{4} \)
$83$ \( 1 + 1693436786 T^{2} + 15516041187205853449 T^{4} \)
$89$ \( ( 1 + 107590 T + 5584059449 T^{2} )^{2} \)
$97$ \( ( 1 + 108838 T + 8587340257 T^{2} )^{2} \)
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