Properties

Label 320.6.a.y
Level $320$
Weight $6$
Character orbit 320.a
Self dual yes
Analytic conductor $51.323$
Analytic rank $0$
Dimension $3$
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: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.39180.1
Defining polynomial: \( x^{3} - x^{2} - 36x - 24 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{6} \)
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 a basis \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_1 + 3) q^{3} + 25 q^{5} + ( - \beta_{2} + \beta_1 + 2) q^{7} + (2 \beta_{2} + 12 \beta_1 + 151) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 + 3) q^{3} + 25 q^{5} + ( - \beta_{2} + \beta_1 + 2) q^{7} + (2 \beta_{2} + 12 \beta_1 + 151) q^{9} + ( - 3 \beta_{2} - 131) q^{11} + (2 \beta_{2} - 20 \beta_1 + 124) q^{13} + (25 \beta_1 + 75) q^{15} + ( - 6 \beta_{2} + 60 \beta_1 + 368) q^{17} + (\beta_{2} + 50 \beta_1 + 1047) q^{19} + (4 \beta_{2} - 72 \beta_1 + 296) q^{21} + ( - 7 \beta_{2} + 43 \beta_1 - 2054) q^{23} + 625 q^{25} + (20 \beta_{2} + 182 \beta_1 + 4534) q^{27} + ( - 52 \beta_{2} + 40 \beta_1 - 138) q^{29} + (42 \beta_{2} + 270 \beta_1 - 1196) q^{31} + (6 \beta_{2} - 380 \beta_1 - 678) q^{33} + ( - 25 \beta_{2} + 25 \beta_1 + 50) q^{35} + (76 \beta_{2} + 200 \beta_1 + 3762) q^{37} + ( - 44 \beta_{2} + 110 \beta_1 - 7138) q^{39} + (22 \beta_{2} + 68 \beta_1 + 4120) q^{41} + ( - 14 \beta_{2} + 221 \beta_1 + 8713) q^{43} + (50 \beta_{2} + 300 \beta_1 + 3775) q^{45} + (25 \beta_{2} + 77 \beta_1 - 12288) q^{47} + ( - 74 \beta_{2} - 316 \beta_1 - 1153) q^{49} + (132 \beta_{2} + 410 \beta_1 + 23634) q^{51} + ( - 114 \beta_{2} + 1140 \beta_1 + 6712) q^{53} + ( - 75 \beta_{2} - 3275) q^{55} + (98 \beta_{2} + 1580 \beta_1 + 22486) q^{57} + ( - 185 \beta_{2} - 70 \beta_1 + 11765) q^{59} + ( - 32 \beta_{2} - 1984 \beta_1 + 8718) q^{61} + (91 \beta_{2} - 263 \beta_1 - 26938) q^{63} + (50 \beta_{2} - 500 \beta_1 + 3100) q^{65} + ( - 82 \beta_{2} - 2345 \beta_1 - 2381) q^{67} + (100 \beta_{2} - 2248 \beta_1 + 9728) q^{69} + ( - 56 \beta_{2} - 1030 \beta_1 - 29002) q^{71} + ( - 150 \beta_{2} - 2340 \beta_1 + 23080) q^{73} + (625 \beta_1 + 1875) q^{75} + ( - 106 \beta_{2} - 860 \beta_1 + 45818) q^{77} + ( - 164 \beta_{2} - 700 \beta_1 + 31272) q^{79} + ( - 162 \beta_{2} + 4916 \beta_1 + 48879) q^{81} + (376 \beta_{2} - 2923 \beta_1 + 10935) q^{83} + ( - 150 \beta_{2} + 1500 \beta_1 + 9200) q^{85} + (184 \beta_{2} - 4094 \beta_1 + 10046) q^{87} + ( - 36 \beta_{2} - 600 \beta_1 + 57774) q^{89} + ( - 106 \beta_{2} + 2110 \beta_1 - 36272) q^{91} + (456 \beta_{2} + 4720 \beta_1 + 104352) q^{93} + (25 \beta_{2} + 1250 \beta_1 + 26175) q^{95} + (430 \beta_{2} - 1420 \beta_1 + 57300) q^{97} + ( - 43 \beta_{2} - 3600 \beta_1 - 115931) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 10 q^{3} + 75 q^{5} + 6 q^{7} + 467 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 10 q^{3} + 75 q^{5} + 6 q^{7} + 467 q^{9} - 396 q^{11} + 354 q^{13} + 250 q^{15} + 1158 q^{17} + 3192 q^{19} + 820 q^{21} - 6126 q^{23} + 1875 q^{25} + 13804 q^{27} - 426 q^{29} - 3276 q^{31} - 2408 q^{33} + 150 q^{35} + 11562 q^{37} - 21348 q^{39} + 12450 q^{41} + 26346 q^{43} + 11675 q^{45} - 36762 q^{47} - 3849 q^{49} + 71444 q^{51} + 21162 q^{53} - 9900 q^{55} + 69136 q^{57} + 35040 q^{59} + 24138 q^{61} - 80986 q^{63} + 8850 q^{65} - 9570 q^{67} + 27036 q^{69} - 88092 q^{71} + 66750 q^{73} + 6250 q^{75} + 136488 q^{77} + 92952 q^{79} + 151391 q^{81} + 30258 q^{83} + 28950 q^{85} + 26228 q^{87} + 172686 q^{89} - 106812 q^{91} + 318232 q^{93} + 79800 q^{95} + 170910 q^{97} - 351436 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - x^{2} - 36x - 24 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 4\nu - 1 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 8\nu^{2} - 16\nu - 189 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta _1 + 1 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{2} + 4\beta _1 + 193 ) / 8 \) 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
−5.11788
−0.688934
6.80681
0 −18.4715 0 25.0000 0 −121.899 0 98.1968 0
1.2 0 −0.755735 0 25.0000 0 172.424 0 −242.429 0
1.3 0 29.2272 0 25.0000 0 −44.5253 0 611.232 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.y 3
4.b odd 2 1 320.6.a.x 3
8.b even 2 1 160.6.a.f 3
8.d odd 2 1 160.6.a.g yes 3
40.e odd 2 1 800.6.a.n 3
40.f even 2 1 800.6.a.o 3
40.i odd 4 2 800.6.c.k 6
40.k even 4 2 800.6.c.j 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
160.6.a.f 3 8.b even 2 1
160.6.a.g yes 3 8.d odd 2 1
320.6.a.x 3 4.b odd 2 1
320.6.a.y 3 1.a even 1 1 trivial
800.6.a.n 3 40.e odd 2 1
800.6.a.o 3 40.f even 2 1
800.6.c.j 6 40.k even 4 2
800.6.c.k 6 40.i odd 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{3} - 10T_{3}^{2} - 548T_{3} - 408 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(320))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \) Copy content Toggle raw display
$3$ \( T^{3} - 10 T^{2} - 548 T - 408 \) Copy content Toggle raw display
$5$ \( (T - 25)^{3} \) Copy content Toggle raw display
$7$ \( T^{3} - 6 T^{2} - 23268 T - 935848 \) Copy content Toggle raw display
$11$ \( T^{3} + 396 T^{2} + \cdots - 59934400 \) Copy content Toggle raw display
$13$ \( T^{3} - 354 T^{2} + \cdots - 28863000 \) Copy content Toggle raw display
$17$ \( T^{3} - 1158 T^{2} + \cdots + 2743838200 \) Copy content Toggle raw display
$19$ \( T^{3} - 3192 T^{2} + \cdots - 126323200 \) Copy content Toggle raw display
$23$ \( T^{3} + 6126 T^{2} + \cdots + 5283148104 \) Copy content Toggle raw display
$29$ \( T^{3} + 426 T^{2} + \cdots - 159249002312 \) Copy content Toggle raw display
$31$ \( T^{3} + 3276 T^{2} + \cdots - 229217617600 \) Copy content Toggle raw display
$37$ \( T^{3} - 11562 T^{2} + \cdots + 1078137168200 \) Copy content Toggle raw display
$41$ \( T^{3} - 12450 T^{2} + \cdots - 1203781400 \) Copy content Toggle raw display
$43$ \( T^{3} - 26346 T^{2} + \cdots - 352818354968 \) Copy content Toggle raw display
$47$ \( T^{3} + 36762 T^{2} + \cdots + 1628197032152 \) Copy content Toggle raw display
$53$ \( T^{3} - 21162 T^{2} + \cdots + 18581204112200 \) Copy content Toggle raw display
$59$ \( T^{3} - 35040 T^{2} + \cdots - 887454720000 \) Copy content Toggle raw display
$61$ \( T^{3} - 24138 T^{2} + \cdots + 47677792189640 \) Copy content Toggle raw display
$67$ \( T^{3} + 9570 T^{2} + \cdots + 57239443872504 \) Copy content Toggle raw display
$71$ \( T^{3} + 88092 T^{2} + \cdots + 11664981864000 \) Copy content Toggle raw display
$73$ \( T^{3} + \cdots + 165394983407000 \) Copy content Toggle raw display
$79$ \( T^{3} - 92952 T^{2} + \cdots - 1808931596800 \) Copy content Toggle raw display
$83$ \( T^{3} + \cdots - 187637358980920 \) Copy content Toggle raw display
$89$ \( T^{3} + \cdots - 175016035497384 \) Copy content Toggle raw display
$97$ \( T^{3} - 170910 T^{2} + \cdots + 83009979455000 \) Copy content Toggle raw display
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