Properties

Label 320.8.c.d
Level $320$
Weight $8$
Character orbit 320.c
Analytic conductor $99.963$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [320,8,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, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("320.129");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 320 = 2^{6} \cdot 5 \)
Weight: \( k \) \(=\) \( 8 \)
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: \(99.9632081549\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-29}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 29 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 5)
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 \(\beta = 2\sqrt{-29}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 3 \beta q^{3} + ( - 25 \beta - 75) q^{5} + 39 \beta q^{7} + 1143 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 3 \beta q^{3} + ( - 25 \beta - 75) q^{5} + 39 \beta q^{7} + 1143 q^{9} + 6828 q^{11} - 942 \beta q^{13} + ( - 225 \beta + 8700) q^{15} - 1456 \beta q^{17} - 6860 q^{19} - 13572 q^{21} - 2713 \beta q^{23} + (3750 \beta - 66875) q^{25} + 9990 \beta q^{27} - 25590 q^{29} + 82112 q^{31} + 20484 \beta q^{33} + ( - 2925 \beta + 113100) q^{35} - 20754 \beta q^{37} + 327816 q^{39} - 533118 q^{41} + 65823 \beta q^{43} + ( - 28575 \beta - 85725) q^{45} - 541 \beta q^{47} + 647107 q^{49} + 506688 q^{51} - 54722 \beta q^{53} + ( - 170700 \beta - 512100) q^{55} - 20580 \beta q^{57} - 1438980 q^{59} - 1381022 q^{61} + 44577 \beta q^{63} + (70650 \beta - 2731800) q^{65} - 252069 \beta q^{67} + 944124 q^{69} - 481608 q^{71} - 137988 \beta q^{73} + ( - 200625 \beta - 1305000) q^{75} + 266292 \beta q^{77} - 1059760 q^{79} - 976779 q^{81} - 241757 \beta q^{83} + (109200 \beta - 4222400) q^{85} - 76770 \beta q^{87} + 5644170 q^{89} + 4261608 q^{91} + 246336 \beta q^{93} + (171500 \beta + 514500) q^{95} - 1115016 \beta q^{97} + 7804404 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 150 q^{5} + 2286 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 150 q^{5} + 2286 q^{9} + 13656 q^{11} + 17400 q^{15} - 13720 q^{19} - 27144 q^{21} - 133750 q^{25} - 51180 q^{29} + 164224 q^{31} + 226200 q^{35} + 655632 q^{39} - 1066236 q^{41} - 171450 q^{45} + 1294214 q^{49} + 1013376 q^{51} - 1024200 q^{55} - 2877960 q^{59} - 2762044 q^{61} - 5463600 q^{65} + 1888248 q^{69} - 963216 q^{71} - 2610000 q^{75} - 2119520 q^{79} - 1953558 q^{81} - 8444800 q^{85} + 11288340 q^{89} + 8523216 q^{91} + 1029000 q^{95} + 15608808 q^{99}+O(q^{100}) \) 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
5.38516i
5.38516i
0 32.3110i 0 −75.0000 + 269.258i 0 420.043i 0 1143.00 0
129.2 0 32.3110i 0 −75.0000 269.258i 0 420.043i 0 1143.00 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
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 320.8.c.d 2
4.b odd 2 1 320.8.c.c 2
5.b even 2 1 inner 320.8.c.d 2
8.b even 2 1 5.8.b.a 2
8.d odd 2 1 80.8.c.a 2
20.d odd 2 1 320.8.c.c 2
24.h odd 2 1 45.8.b.a 2
40.e odd 2 1 80.8.c.a 2
40.f even 2 1 5.8.b.a 2
40.i odd 4 2 25.8.a.d 2
40.k even 4 2 400.8.a.y 2
120.i odd 2 1 45.8.b.a 2
120.w even 4 2 225.8.a.n 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5.8.b.a 2 8.b even 2 1
5.8.b.a 2 40.f even 2 1
25.8.a.d 2 40.i odd 4 2
45.8.b.a 2 24.h odd 2 1
45.8.b.a 2 120.i odd 2 1
80.8.c.a 2 8.d odd 2 1
80.8.c.a 2 40.e odd 2 1
225.8.a.n 2 120.w even 4 2
320.8.c.c 2 4.b odd 2 1
320.8.c.c 2 20.d odd 2 1
320.8.c.d 2 1.a even 1 1 trivial
320.8.c.d 2 5.b even 2 1 inner
400.8.a.y 2 40.k even 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{8}^{\mathrm{new}}(320, [\chi])\):

\( T_{3}^{2} + 1044 \) Copy content Toggle raw display
\( T_{11} - 6828 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 1044 \) Copy content Toggle raw display
$5$ \( T^{2} + 150T + 78125 \) Copy content Toggle raw display
$7$ \( T^{2} + 176436 \) Copy content Toggle raw display
$11$ \( (T - 6828)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 102934224 \) Copy content Toggle raw display
$17$ \( T^{2} + 245912576 \) Copy content Toggle raw display
$19$ \( (T + 6860)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 853802804 \) Copy content Toggle raw display
$29$ \( (T + 25590)^{2} \) Copy content Toggle raw display
$31$ \( (T - 82112)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 49964507856 \) Copy content Toggle raw display
$41$ \( (T + 533118)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 502589410164 \) Copy content Toggle raw display
$47$ \( T^{2} + 33950996 \) Copy content Toggle raw display
$53$ \( T^{2} + 347361684944 \) Copy content Toggle raw display
$59$ \( (T + 1438980)^{2} \) Copy content Toggle raw display
$61$ \( (T + 1381022)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 7370498568276 \) Copy content Toggle raw display
$71$ \( (T + 481608)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 2208719824704 \) Copy content Toggle raw display
$79$ \( (T + 1059760)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 6779787857684 \) Copy content Toggle raw display
$89$ \( (T - 5644170)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + \cdots + 144218238909696 \) Copy content Toggle raw display
show more
show less