Properties

Label 320.4.c.i
Level $320$
Weight $4$
Character orbit 320.c
Analytic conductor $18.881$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [320,4,Mod(129,320)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(320, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("320.129");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 320 = 2^{6} \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 320.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(18.8806112018\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{29})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 15x^{2} + 49 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{8} \)
Twist minimal: no (minimal twist has level 160)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

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.

\(f(q)\) \(=\) \( q - \beta_{2} q^{3} + ( - \beta_1 + 3) q^{5} + \beta_{2} q^{7} + 11 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{2} q^{3} + ( - \beta_1 + 3) q^{5} + \beta_{2} q^{7} + 11 q^{9} - \beta_{3} q^{11} - 2 \beta_1 q^{13} + (\beta_{3} - 3 \beta_{2}) q^{15} - 4 \beta_1 q^{17} - 3 \beta_{3} q^{19} + 16 q^{21} + 13 \beta_{2} q^{23} + ( - 6 \beta_1 - 107) q^{25} - 38 \beta_{2} q^{27} - 158 q^{29} + 4 \beta_{3} q^{31} - 16 \beta_1 q^{33} + ( - \beta_{3} + 3 \beta_{2}) q^{35} - 26 \beta_1 q^{37} + 2 \beta_{3} q^{39} - 170 q^{41} + 79 \beta_{2} q^{43} + ( - 11 \beta_1 + 33) q^{45} + 61 \beta_{2} q^{47} + 327 q^{49} + 4 \beta_{3} q^{51} - 46 \beta_1 q^{53} + ( - 3 \beta_{3} - 116 \beta_{2}) q^{55} - 48 \beta_1 q^{57} + 15 \beta_{3} q^{59} - 82 q^{61} + 11 \beta_{2} q^{63} + ( - 6 \beta_1 - 232) q^{65} - 173 \beta_{2} q^{67} + 208 q^{69} - 22 \beta_{3} q^{71} - 40 \beta_1 q^{73} + (6 \beta_{3} + 107 \beta_{2}) q^{75} + 16 \beta_1 q^{77} - 8 \beta_{3} q^{79} - 311 q^{81} + 235 \beta_{2} q^{83} + ( - 12 \beta_1 - 464) q^{85} + 158 \beta_{2} q^{87} - 6 q^{89} - 2 \beta_{3} q^{91} + 64 \beta_1 q^{93} + ( - 9 \beta_{3} - 348 \beta_{2}) q^{95} + 100 \beta_1 q^{97} - 11 \beta_{3} q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 12 q^{5} + 44 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 12 q^{5} + 44 q^{9} + 64 q^{21} - 428 q^{25} - 632 q^{29} - 680 q^{41} + 132 q^{45} + 1308 q^{49} - 328 q^{61} - 928 q^{65} + 832 q^{69} - 1244 q^{81} - 1856 q^{85} - 24 q^{89}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 15x^{2} + 49 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 2\nu^{3} + 44\nu ) / 7 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -4\nu^{3} - 32\nu ) / 7 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 16\nu^{2} + 120 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} + 2\beta_1 ) / 8 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} - 120 ) / 16 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -11\beta_{2} - 8\beta_1 ) / 4 \) Copy content Toggle raw display

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
129.1
3.19258i
2.19258i
2.19258i
3.19258i
0 4.00000i 0 3.00000 10.7703i 0 4.00000i 0 11.0000 0
129.2 0 4.00000i 0 3.00000 + 10.7703i 0 4.00000i 0 11.0000 0
129.3 0 4.00000i 0 3.00000 10.7703i 0 4.00000i 0 11.0000 0
129.4 0 4.00000i 0 3.00000 + 10.7703i 0 4.00000i 0 11.0000 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
5.b even 2 1 inner
20.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 320.4.c.i 4
4.b odd 2 1 inner 320.4.c.i 4
5.b even 2 1 inner 320.4.c.i 4
5.c odd 4 1 1600.4.a.cc 2
5.c odd 4 1 1600.4.a.co 2
8.b even 2 1 160.4.c.b 4
8.d odd 2 1 160.4.c.b 4
20.d odd 2 1 inner 320.4.c.i 4
20.e even 4 1 1600.4.a.cc 2
20.e even 4 1 1600.4.a.co 2
24.f even 2 1 1440.4.f.h 4
24.h odd 2 1 1440.4.f.h 4
40.e odd 2 1 160.4.c.b 4
40.f even 2 1 160.4.c.b 4
40.i odd 4 1 800.4.a.n 2
40.i odd 4 1 800.4.a.r 2
40.k even 4 1 800.4.a.n 2
40.k even 4 1 800.4.a.r 2
120.i odd 2 1 1440.4.f.h 4
120.m even 2 1 1440.4.f.h 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
160.4.c.b 4 8.b even 2 1
160.4.c.b 4 8.d odd 2 1
160.4.c.b 4 40.e odd 2 1
160.4.c.b 4 40.f even 2 1
320.4.c.i 4 1.a even 1 1 trivial
320.4.c.i 4 4.b odd 2 1 inner
320.4.c.i 4 5.b even 2 1 inner
320.4.c.i 4 20.d odd 2 1 inner
800.4.a.n 2 40.i odd 4 1
800.4.a.n 2 40.k even 4 1
800.4.a.r 2 40.i odd 4 1
800.4.a.r 2 40.k even 4 1
1440.4.f.h 4 24.f even 2 1
1440.4.f.h 4 24.h odd 2 1
1440.4.f.h 4 120.i odd 2 1
1440.4.f.h 4 120.m even 2 1
1600.4.a.cc 2 5.c odd 4 1
1600.4.a.cc 2 20.e even 4 1
1600.4.a.co 2 5.c odd 4 1
1600.4.a.co 2 20.e even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(320, [\chi])\):

\( T_{3}^{2} + 16 \) Copy content Toggle raw display
\( T_{11}^{2} - 1856 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( (T^{2} + 16)^{2} \) Copy content Toggle raw display
$5$ \( (T^{2} - 6 T + 125)^{2} \) Copy content Toggle raw display
$7$ \( (T^{2} + 16)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} - 1856)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 464)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} + 1856)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} - 16704)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} + 2704)^{2} \) Copy content Toggle raw display
$29$ \( (T + 158)^{4} \) Copy content Toggle raw display
$31$ \( (T^{2} - 29696)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 78416)^{2} \) Copy content Toggle raw display
$41$ \( (T + 170)^{4} \) Copy content Toggle raw display
$43$ \( (T^{2} + 99856)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} + 59536)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} + 245456)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} - 417600)^{2} \) Copy content Toggle raw display
$61$ \( (T + 82)^{4} \) Copy content Toggle raw display
$67$ \( (T^{2} + 478864)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} - 898304)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} + 185600)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} - 118784)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} + 883600)^{2} \) Copy content Toggle raw display
$89$ \( (T + 6)^{4} \) Copy content Toggle raw display
$97$ \( (T^{2} + 1160000)^{2} \) Copy content Toggle raw display
show more
show less