Properties

Label 320.6.c.h
Level 320
Weight 6
Character orbit 320.c
Analytic conductor 51.323
Analytic rank 0
Dimension 4
CM discriminant -20
Inner twists 4

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) \(=\) \( 320 = 2^{6} \cdot 5 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 320.c (of order \(2\), degree \(1\), not 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: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4}\cdot 5^{2} \)
Twist minimal: no (minimal twist has level 160)
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.

\(f(q)\) \(=\) \( q + ( 5 \beta_{1} + \beta_{2} ) q^{3} -5 \beta_{3} q^{5} + ( -46 \beta_{1} - \beta_{2} ) q^{7} + ( -243 + 38 \beta_{3} ) q^{9} +O(q^{10})\) \( q + ( 5 \beta_{1} + \beta_{2} ) q^{3} -5 \beta_{3} q^{5} + ( -46 \beta_{1} - \beta_{2} ) q^{7} + ( -243 + 38 \beta_{3} ) q^{9} + ( 180 \beta_{1} + 95 \beta_{2} ) q^{15} + ( 3602 - 202 \beta_{3} ) q^{21} + ( -664 \beta_{1} - 237 \beta_{2} ) q^{23} + 3125 q^{25} + ( -1368 \beta_{1} - 722 \beta_{2} ) q^{27} -1686 q^{29} + ( -385 \beta_{1} - 915 \beta_{2} ) q^{35} -1882 \beta_{3} q^{41} + ( 1123 \beta_{1} - 1439 \beta_{2} ) q^{43} + ( -23750 + 1215 \beta_{3} ) q^{45} + ( -3866 \beta_{1} + 1203 \beta_{2} ) q^{47} + ( -16807 + 366 \beta_{3} ) q^{49} -4674 \beta_{3} q^{61} + ( 14104 \beta_{1} + 7197 \beta_{2} ) q^{63} + ( -12853 \beta_{1} - 1195 \beta_{2} ) q^{67} + ( 75586 - 6922 \beta_{3} ) q^{69} + ( 15625 \beta_{1} + 3125 \beta_{2} ) q^{75} + ( 121451 - 9234 \beta_{3} ) q^{81} + ( 21187 \beta_{1} + 3117 \beta_{2} ) q^{83} + ( -8430 \beta_{1} - 1686 \beta_{2} ) q^{87} -149286 q^{89} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 972q^{9} + O(q^{10}) \) \( 4q - 972q^{9} + 14408q^{21} + 12500q^{25} - 6744q^{29} - 95000q^{45} - 67228q^{49} + 302344q^{69} + 485804q^{81} - 597144q^{89} + O(q^{100}) \)

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

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( -4 \nu^{3} - 8 \nu \)
\(\beta_{2}\)\(=\)\( 6 \nu^{3} + 22 \nu \)
\(\beta_{3}\)\(=\)\( 10 \nu^{2} + 15 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(2 \beta_{2} + 3 \beta_{1}\)\()/20\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{3} - 15\)\()/10\)
\(\nu^{3}\)\(=\)\((\)\(-4 \beta_{2} - 11 \beta_{1}\)\()/20\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
129.1
1.61803i
0.618034i
0.618034i
1.61803i
0 30.1803i 0 55.9017 0 194.180i 0 −667.853 0
129.2 0 7.81966i 0 −55.9017 0 171.820i 0 181.853 0
129.3 0 7.81966i 0 −55.9017 0 171.820i 0 181.853 0
129.4 0 30.1803i 0 55.9017 0 194.180i 0 −667.853 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
20.d odd 2 1 CM by \(\Q(\sqrt{-5}) \)
4.b odd 2 1 inner
5.b 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.c.h 4
4.b odd 2 1 inner 320.6.c.h 4
5.b even 2 1 inner 320.6.c.h 4
8.b even 2 1 160.6.c.b 4
8.d odd 2 1 160.6.c.b 4
20.d odd 2 1 CM 320.6.c.h 4
40.e odd 2 1 160.6.c.b 4
40.f even 2 1 160.6.c.b 4
40.i odd 4 1 800.6.a.f 2
40.i odd 4 1 800.6.a.m 2
40.k even 4 1 800.6.a.f 2
40.k even 4 1 800.6.a.m 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
160.6.c.b 4 8.b even 2 1
160.6.c.b 4 8.d odd 2 1
160.6.c.b 4 40.e odd 2 1
160.6.c.b 4 40.f even 2 1
320.6.c.h 4 1.a even 1 1 trivial
320.6.c.h 4 4.b odd 2 1 inner
320.6.c.h 4 5.b even 2 1 inner
320.6.c.h 4 20.d odd 2 1 CM
800.6.a.f 2 40.i odd 4 1
800.6.a.f 2 40.k even 4 1
800.6.a.m 2 40.i odd 4 1
800.6.a.m 2 40.k even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{6}^{\mathrm{new}}(320, [\chi])\):

\( T_{3}^{4} + 972 T_{3}^{2} + 55696 \)
\( T_{11} \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( ( 1 - 38 T + 722 T^{2} - 9234 T^{3} + 59049 T^{4} )( 1 + 38 T + 722 T^{2} + 9234 T^{3} + 59049 T^{4} ) \)
$5$ \( ( 1 - 3125 T^{2} )^{2} \)
$7$ \( ( 1 - 366 T + 66978 T^{2} - 6151362 T^{3} + 282475249 T^{4} )( 1 + 366 T + 66978 T^{2} + 6151362 T^{3} + 282475249 T^{4} ) \)
$11$ \( ( 1 + 161051 T^{2} )^{4} \)
$13$ \( ( 1 - 371293 T^{2} )^{4} \)
$17$ \( ( 1 - 1419857 T^{2} )^{4} \)
$19$ \( ( 1 + 2476099 T^{2} )^{4} \)
$23$ \( ( 1 - 4838 T + 11703122 T^{2} - 31139027434 T^{3} + 41426511213649 T^{4} )( 1 + 4838 T + 11703122 T^{2} + 31139027434 T^{3} + 41426511213649 T^{4} ) \)
$29$ \( ( 1 + 1686 T + 20511149 T^{2} )^{4} \)
$31$ \( ( 1 + 28629151 T^{2} )^{4} \)
$37$ \( ( 1 - 69343957 T^{2} )^{4} \)
$41$ \( ( 1 - 211028098 T^{2} + 13422659310152401 T^{4} )^{2} \)
$43$ \( ( 1 - 11862 T + 70353522 T^{2} - 1743814150866 T^{3} + 21611482313284249 T^{4} )( 1 + 11862 T + 70353522 T^{2} + 1743814150866 T^{3} + 21611482313284249 T^{4} ) \)
$47$ \( ( 1 - 33334 T + 555577778 T^{2} - 7644986463338 T^{3} + 52599132235830049 T^{4} )( 1 + 33334 T + 555577778 T^{2} + 7644986463338 T^{3} + 52599132235830049 T^{4} ) \)
$53$ \( ( 1 - 418195493 T^{2} )^{4} \)
$59$ \( ( 1 + 714924299 T^{2} )^{4} \)
$61$ \( ( 1 - 1041591898 T^{2} + 713342911662882601 T^{4} )^{2} \)
$67$ \( ( 1 - 100434 T + 5043494178 T^{2} - 135598464996438 T^{3} + 1822837804551761449 T^{4} )( 1 + 100434 T + 5043494178 T^{2} + 135598464996438 T^{3} + 1822837804551761449 T^{4} ) \)
$71$ \( ( 1 + 1804229351 T^{2} )^{4} \)
$73$ \( ( 1 - 2073071593 T^{2} )^{4} \)
$79$ \( ( 1 + 3077056399 T^{2} )^{4} \)
$83$ \( ( 1 - 163262 T + 13327240322 T^{2} - 643095653457466 T^{3} + 15516041187205853449 T^{4} )( 1 + 163262 T + 13327240322 T^{2} + 643095653457466 T^{3} + 15516041187205853449 T^{4} ) \)
$89$ \( ( 1 + 149286 T + 5584059449 T^{2} )^{4} \)
$97$ \( ( 1 - 8587340257 T^{2} )^{4} \)
show more
show less