Properties

Label 320.6.f.a
Level 320
Weight 6
Character orbit 320.f
Analytic conductor 51.323
Analytic rank 0
Dimension 4
CM discriminant -40
Inner twists 8

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Newspace parameters

Level: \( N \) \(=\) \( 320 = 2^{6} \cdot 5 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 320.f (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(51.3228223402\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{5})\)
Defining polynomial: \(x^{4} + 3 x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{4}\cdot 5^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 5 \beta_{3} q^{5} -11 \beta_{2} q^{7} -243 q^{9} +O(q^{10})\) \( q + 5 \beta_{3} q^{5} -11 \beta_{2} q^{7} -243 q^{9} + 401 \beta_{1} q^{11} -22 \beta_{3} q^{13} + 957 \beta_{1} q^{19} + 149 \beta_{2} q^{23} + 3125 q^{25} -6875 \beta_{1} q^{35} + 1418 \beta_{3} q^{37} + 15202 q^{41} -1215 \beta_{3} q^{45} -687 \beta_{2} q^{47} -43693 q^{49} + 1506 \beta_{3} q^{53} + 2005 \beta_{2} q^{55} + 13993 \beta_{1} q^{59} + 2673 \beta_{2} q^{63} -13750 q^{65} + 17644 \beta_{3} q^{77} + 59049 q^{81} + 128786 q^{89} + 30250 \beta_{1} q^{91} + 4785 \beta_{2} q^{95} -97443 \beta_{1} q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 972q^{9} + O(q^{10}) \) \( 4q - 972q^{9} + 12500q^{25} + 60808q^{41} - 174772q^{49} - 55000q^{65} + 236196q^{81} + 515144q^{89} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 3 x^{2} + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 2 \nu^{3} + 4 \nu \)
\(\beta_{2}\)\(=\)\( 10 \nu^{3} + 40 \nu \)
\(\beta_{3}\)\(=\)\( 10 \nu^{2} + 15 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{2} - 5 \beta_{1}\)\()/20\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{3} - 15\)\()/10\)
\(\nu^{3}\)\(=\)\((\)\(-\beta_{2} + 10 \beta_{1}\)\()/10\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/320\mathbb{Z}\right)^\times\).

\(n\) \(191\) \(257\) \(261\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
289.1
1.61803i
1.61803i
0.618034i
0.618034i
0 0 0 −55.9017 0 245.967i 0 −243.000 0
289.2 0 0 0 −55.9017 0 245.967i 0 −243.000 0
289.3 0 0 0 55.9017 0 245.967i 0 −243.000 0
289.4 0 0 0 55.9017 0 245.967i 0 −243.000 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
40.e odd 2 1 CM by \(\Q(\sqrt{-10}) \)
4.b odd 2 1 inner
5.b even 2 1 inner
8.b even 2 1 inner
8.d odd 2 1 inner
20.d odd 2 1 inner
40.f even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 320.6.f.a 4
4.b odd 2 1 inner 320.6.f.a 4
5.b even 2 1 inner 320.6.f.a 4
8.b even 2 1 inner 320.6.f.a 4
8.d odd 2 1 inner 320.6.f.a 4
20.d odd 2 1 inner 320.6.f.a 4
40.e odd 2 1 CM 320.6.f.a 4
40.f even 2 1 inner 320.6.f.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
320.6.f.a 4 1.a even 1 1 trivial
320.6.f.a 4 4.b odd 2 1 inner
320.6.f.a 4 5.b even 2 1 inner
320.6.f.a 4 8.b even 2 1 inner
320.6.f.a 4 8.d odd 2 1 inner
320.6.f.a 4 20.d odd 2 1 inner
320.6.f.a 4 40.e odd 2 1 CM
320.6.f.a 4 40.f even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} \) acting on \(S_{6}^{\mathrm{new}}(320, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( ( 1 + 243 T^{2} )^{4} \)
$5$ \( ( 1 - 3125 T^{2} )^{2} \)
$7$ \( ( 1 + 26886 T^{2} + 282475249 T^{4} )^{2} \)
$11$ \( ( 1 + 321102 T^{2} + 25937424601 T^{4} )^{2} \)
$13$ \( ( 1 + 682086 T^{2} + 137858491849 T^{4} )^{2} \)
$17$ \( ( 1 - 1419857 T^{2} )^{4} \)
$19$ \( ( 1 - 1288802 T^{2} + 6131066257801 T^{4} )^{2} \)
$23$ \( ( 1 - 1772186 T^{2} + 41426511213649 T^{4} )^{2} \)
$29$ \( ( 1 - 20511149 T^{2} )^{4} \)
$31$ \( ( 1 + 28629151 T^{2} )^{4} \)
$37$ \( ( 1 - 112652586 T^{2} + 4808584372417849 T^{4} )^{2} \)
$41$ \( ( 1 - 15202 T + 115856201 T^{2} )^{4} \)
$43$ \( ( 1 + 147008443 T^{2} )^{4} \)
$47$ \( ( 1 - 222705514 T^{2} + 52599132235830049 T^{4} )^{2} \)
$53$ \( ( 1 + 552886486 T^{2} + 174887470365513049 T^{4} )^{2} \)
$59$ \( ( 1 - 646632402 T^{2} + 511116753300641401 T^{4} )^{2} \)
$61$ \( ( 1 - 844596301 T^{2} )^{4} \)
$67$ \( ( 1 + 1350125107 T^{2} )^{4} \)
$71$ \( ( 1 + 1804229351 T^{2} )^{4} \)
$73$ \( ( 1 - 2073071593 T^{2} )^{4} \)
$79$ \( ( 1 + 3077056399 T^{2} )^{4} \)
$83$ \( ( 1 + 3939040643 T^{2} )^{4} \)
$89$ \( ( 1 - 128786 T + 5584059449 T^{2} )^{4} \)
$97$ \( ( 1 - 8587340257 T^{2} )^{4} \)
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