Properties

Label 320.6.f.b
Level 320
Weight 6
Character orbit 320.f
Analytic conductor 51.323
Analytic rank 0
Dimension 8
CM no
Inner twists 8

Related objects

Downloads

Learn more about

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: \(8\)
Coefficient field: 8.0.73499483897856.45
Defining polynomial: \(x^{8} - 3721 x^{4} + 13845841\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{10} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{4} q^{3} -5 \beta_{5} q^{5} -11 \beta_{2} q^{7} + 123 q^{9} +O(q^{10})\) \( q + \beta_{4} q^{3} -5 \beta_{5} q^{5} -11 \beta_{2} q^{7} + 123 q^{9} -149 \beta_{1} q^{11} + ( 44 \beta_{5} + 22 \beta_{6} ) q^{13} + ( 15 \beta_{2} - 305 \beta_{3} ) q^{15} -11 \beta_{7} q^{17} -1287 \beta_{1} q^{19} -671 \beta_{6} q^{21} -65 \beta_{2} q^{23} + ( 2975 + 25 \beta_{7} ) q^{25} -120 \beta_{4} q^{27} -1078 \beta_{6} q^{29} + 1798 \beta_{3} q^{31} -149 \beta_{7} q^{33} + ( -3355 \beta_{1} + 55 \beta_{4} ) q^{35} + ( 518 \beta_{5} + 259 \beta_{6} ) q^{37} + 2684 \beta_{3} q^{39} + 11396 q^{41} + 1001 \beta_{4} q^{43} -615 \beta_{5} q^{45} + 207 \beta_{2} q^{47} + 2045 q^{49} -4026 \beta_{1} q^{51} + ( 1560 \beta_{5} + 780 \beta_{6} ) q^{53} + ( -1490 \beta_{2} - 745 \beta_{3} ) q^{55} -1287 \beta_{7} q^{57} + 8189 \beta_{1} q^{59} + 14069 \beta_{6} q^{61} -1353 \beta_{2} q^{63} + ( -26840 - 110 \beta_{7} ) q^{65} + 439 \beta_{4} q^{67} -3965 \beta_{6} q^{69} -8398 \beta_{3} q^{71} -11 \beta_{7} q^{73} + ( 9150 \beta_{1} + 2975 \beta_{4} ) q^{75} + ( 3278 \beta_{5} + 1639 \beta_{6} ) q^{77} -8998 \beta_{3} q^{79} -73809 q^{81} + 5555 \beta_{4} q^{83} + ( -330 \beta_{5} - 6875 \beta_{6} ) q^{85} -6468 \beta_{2} q^{87} -99362 q^{89} + 29524 \beta_{1} q^{91} + ( 10788 \beta_{5} + 5394 \beta_{6} ) q^{93} + ( -12870 \beta_{2} - 6435 \beta_{3} ) q^{95} -4713 \beta_{7} q^{97} -18327 \beta_{1} q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 984q^{9} + O(q^{10}) \) \( 8q + 984q^{9} + 23800q^{25} + 91168q^{41} + 16360q^{49} - 214720q^{65} - 590472q^{81} - 794896q^{89} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - 3721 x^{4} + 13845841\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 2 \nu^{6} \)\(/226981\)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{5} - 61 \nu^{3} + 3721 \nu \)\()/3721\)
\(\beta_{3}\)\(=\)\((\)\( -2 \nu^{6} + 14884 \nu^{2} \)\()/226981\)
\(\beta_{4}\)\(=\)\((\)\( -2 \nu^{7} + 61 \nu^{5} + 3721 \nu^{3} + 226981 \nu \)\()/226981\)
\(\beta_{5}\)\(=\)\((\)\( -\nu^{5} + 2 \nu^{4} + 61 \nu^{3} + 3721 \nu - 3721 \)\()/3721\)
\(\beta_{6}\)\(=\)\((\)\( -4 \nu^{4} + 7442 \)\()/3721\)
\(\beta_{7}\)\(=\)\((\)\( 4 \nu^{7} + 122 \nu^{5} - 7442 \nu^{3} + 453962 \nu \)\()/226981\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{7} + \beta_{6} + 2 \beta_{5} + 2 \beta_{4} + 2 \beta_{2}\)\()/8\)
\(\nu^{2}\)\(=\)\((\)\(61 \beta_{3} + 61 \beta_{1}\)\()/4\)
\(\nu^{3}\)\(=\)\((\)\(61 \beta_{6} + 122 \beta_{5} - 122 \beta_{2}\)\()/4\)
\(\nu^{4}\)\(=\)\((\)\(-3721 \beta_{6} + 7442\)\()/4\)
\(\nu^{5}\)\(=\)\((\)\(3721 \beta_{7} - 3721 \beta_{6} - 7442 \beta_{5} + 7442 \beta_{4} - 7442 \beta_{2}\)\()/8\)
\(\nu^{6}\)\(=\)\(226981 \beta_{1}\)\(/2\)
\(\nu^{7}\)\(=\)\((\)\(226981 \beta_{7} + 226981 \beta_{6} + 453962 \beta_{5} - 453962 \beta_{4} - 453962 \beta_{2}\)\()/8\)

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
−2.02144 + 7.54412i
−2.02144 7.54412i
−7.54412 2.02144i
−7.54412 + 2.02144i
7.54412 + 2.02144i
7.54412 2.02144i
2.02144 7.54412i
2.02144 + 7.54412i
0 −19.1311 0 −55.2268 8.66025i 0 121.499i 0 123.000 0
289.2 0 −19.1311 0 −55.2268 + 8.66025i 0 121.499i 0 123.000 0
289.3 0 −19.1311 0 55.2268 8.66025i 0 121.499i 0 123.000 0
289.4 0 −19.1311 0 55.2268 + 8.66025i 0 121.499i 0 123.000 0
289.5 0 19.1311 0 −55.2268 8.66025i 0 121.499i 0 123.000 0
289.6 0 19.1311 0 −55.2268 + 8.66025i 0 121.499i 0 123.000 0
289.7 0 19.1311 0 55.2268 8.66025i 0 121.499i 0 123.000 0
289.8 0 19.1311 0 55.2268 + 8.66025i 0 121.499i 0 123.000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 289.8
Significant digits:
Format:

Inner twists

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} - 366 \) acting on \(S_{6}^{\mathrm{new}}(320, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( ( 1 + 120 T^{2} + 59049 T^{4} )^{4} \)
$5$ \( ( 1 - 5950 T^{2} + 9765625 T^{4} )^{2} \)
$7$ \( ( 1 - 18852 T^{2} + 282475249 T^{4} )^{4} \)
$11$ \( ( 1 - 233298 T^{2} + 25937424601 T^{4} )^{4} \)
$13$ \( ( 1 + 506394 T^{2} + 137858491849 T^{4} )^{4} \)
$17$ \( ( 1 - 2662570 T^{2} + 2015993900449 T^{4} )^{4} \)
$19$ \( ( 1 + 1673278 T^{2} + 6131066257801 T^{4} )^{4} \)
$23$ \( ( 1 - 12357236 T^{2} + 41426511213649 T^{4} )^{4} \)
$29$ \( ( 1 - 27077290 T^{2} + 420707233300201 T^{4} )^{4} \)
$31$ \( ( 1 + 18464654 T^{2} + 819628286980801 T^{4} )^{4} \)
$37$ \( ( 1 + 105952386 T^{2} + 4808584372417849 T^{4} )^{4} \)
$41$ \( ( 1 - 11396 T + 115856201 T^{2} )^{8} \)
$43$ \( ( 1 - 72715480 T^{2} + 21611482313284249 T^{4} )^{4} \)
$47$ \( ( 1 - 453462436 T^{2} + 52599132235830049 T^{4} )^{4} \)
$53$ \( ( 1 + 539491786 T^{2} + 174887470365513049 T^{4} )^{4} \)
$59$ \( ( 1 - 1161609714 T^{2} + 511116753300641401 T^{4} )^{4} \)
$61$ \( ( 1 + 686048530 T^{2} + 713342911662882601 T^{4} )^{4} \)
$67$ \( ( 1 + 2629714328 T^{2} + 1822837804551761449 T^{4} )^{4} \)
$71$ \( ( 1 + 2762141854 T^{2} + 3255243551009881201 T^{4} )^{4} \)
$73$ \( ( 1 - 4145966042 T^{2} + 4297625829703557649 T^{4} )^{4} \)
$79$ \( ( 1 + 5182544750 T^{2} + 9468276082626847201 T^{4} )^{4} \)
$83$ \( ( 1 - 3415955864 T^{2} + 15516041187205853449 T^{4} )^{4} \)
$89$ \( ( 1 + 99362 T + 5584059449 T^{2} )^{8} \)
$97$ \( ( 1 + 15344227702 T^{2} + 73742412689492826049 T^{4} )^{4} \)
show more
show less