Properties

Label 320.6.c.g
Level $320$
Weight $6$
Character orbit 320.c
Analytic conductor $51.323$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [320,6,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, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("320.129");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 320 = 2^{6} \cdot 5 \)
Weight: \( k \) \(=\) \( 6 \)
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: \(51.3228223402\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-11}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
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 \(\beta = 2\sqrt{-11}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 3 \beta q^{3} + ( - 5 \beta + 45) q^{5} + 9 \beta q^{7} - 153 q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - 3 \beta q^{3} + ( - 5 \beta + 45) q^{5} + 9 \beta q^{7} - 153 q^{9} + 252 q^{11} + 18 \beta q^{13} + ( - 135 \beta - 660) q^{15} - 104 \beta q^{17} - 220 q^{19} + 1188 q^{21} - 367 \beta q^{23} + ( - 450 \beta + 925) q^{25} - 270 \beta q^{27} + 6930 q^{29} - 6752 q^{31} - 756 \beta q^{33} + (405 \beta + 1980) q^{35} - 2106 \beta q^{37} + 2376 q^{39} - 198 q^{41} - 63 \beta q^{43} + (765 \beta - 6885) q^{45} + 1589 \beta q^{47} + 13243 q^{49} - 13728 q^{51} + 878 \beta q^{53} + ( - 1260 \beta + 11340) q^{55} + 660 \beta q^{57} - 24660 q^{59} + 5698 q^{61} - 1377 \beta q^{63} + (810 \beta + 3960) q^{65} - 6579 \beta q^{67} - 48444 q^{69} - 53352 q^{71} + 10692 \beta q^{73} + ( - 2775 \beta - 59400) q^{75} + 2268 \beta q^{77} - 51920 q^{79} - 72819 q^{81} - 9323 \beta q^{83} + ( - 4680 \beta - 22880) q^{85} - 20790 \beta q^{87} - 9990 q^{89} - 7128 q^{91} + 20256 \beta q^{93} + (1100 \beta - 9900) q^{95} - 15264 \beta q^{97} - 38556 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 90 q^{5} - 306 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 90 q^{5} - 306 q^{9} + 504 q^{11} - 1320 q^{15} - 440 q^{19} + 2376 q^{21} + 1850 q^{25} + 13860 q^{29} - 13504 q^{31} + 3960 q^{35} + 4752 q^{39} - 396 q^{41} - 13770 q^{45} + 26486 q^{49} - 27456 q^{51} + 22680 q^{55} - 49320 q^{59} + 11396 q^{61} + 7920 q^{65} - 96888 q^{69} - 106704 q^{71} - 118800 q^{75} - 103840 q^{79} - 145638 q^{81} - 45760 q^{85} - 19980 q^{89} - 14256 q^{91} - 19800 q^{95} - 77112 q^{99}+O(q^{100}) \) 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
0.500000 + 1.65831i
0.500000 1.65831i
0 19.8997i 0 45.0000 33.1662i 0 59.6992i 0 −153.000 0
129.2 0 19.8997i 0 45.0000 + 33.1662i 0 59.6992i 0 −153.000 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
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.g 2
4.b odd 2 1 320.6.c.f 2
5.b even 2 1 inner 320.6.c.g 2
8.b even 2 1 80.6.c.a 2
8.d odd 2 1 5.6.b.a 2
20.d odd 2 1 320.6.c.f 2
24.f even 2 1 45.6.b.b 2
24.h odd 2 1 720.6.f.f 2
40.e odd 2 1 5.6.b.a 2
40.f even 2 1 80.6.c.a 2
40.i odd 4 2 400.6.a.t 2
40.k even 4 2 25.6.a.c 2
56.e even 2 1 245.6.b.a 2
120.i odd 2 1 720.6.f.f 2
120.m even 2 1 45.6.b.b 2
120.q odd 4 2 225.6.a.n 2
280.n even 2 1 245.6.b.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5.6.b.a 2 8.d odd 2 1
5.6.b.a 2 40.e odd 2 1
25.6.a.c 2 40.k even 4 2
45.6.b.b 2 24.f even 2 1
45.6.b.b 2 120.m even 2 1
80.6.c.a 2 8.b even 2 1
80.6.c.a 2 40.f even 2 1
225.6.a.n 2 120.q odd 4 2
245.6.b.a 2 56.e even 2 1
245.6.b.a 2 280.n even 2 1
320.6.c.f 2 4.b odd 2 1
320.6.c.f 2 20.d odd 2 1
320.6.c.g 2 1.a even 1 1 trivial
320.6.c.g 2 5.b even 2 1 inner
400.6.a.t 2 40.i odd 4 2
720.6.f.f 2 24.h odd 2 1
720.6.f.f 2 120.i odd 2 1

Hecke kernels

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

\( T_{3}^{2} + 396 \) Copy content Toggle raw display
\( T_{11} - 252 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 396 \) Copy content Toggle raw display
$5$ \( T^{2} - 90T + 3125 \) Copy content Toggle raw display
$7$ \( T^{2} + 3564 \) Copy content Toggle raw display
$11$ \( (T - 252)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 14256 \) Copy content Toggle raw display
$17$ \( T^{2} + 475904 \) Copy content Toggle raw display
$19$ \( (T + 220)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 5926316 \) Copy content Toggle raw display
$29$ \( (T - 6930)^{2} \) Copy content Toggle raw display
$31$ \( (T + 6752)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 195150384 \) Copy content Toggle raw display
$41$ \( (T + 198)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 174636 \) Copy content Toggle raw display
$47$ \( T^{2} + 111096524 \) Copy content Toggle raw display
$53$ \( T^{2} + 33918896 \) Copy content Toggle raw display
$59$ \( (T + 24660)^{2} \) Copy content Toggle raw display
$61$ \( (T - 5698)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 1904462604 \) Copy content Toggle raw display
$71$ \( (T + 53352)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 5030030016 \) Copy content Toggle raw display
$79$ \( (T + 51920)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 3824406476 \) Copy content Toggle raw display
$89$ \( (T + 9990)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 10251546624 \) Copy content Toggle raw display
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