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Properties

Label 320.3.m.a
Level $320$
Weight $3$
Character orbit 320.m
Analytic conductor $8.719$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [320,3,Mod(33,320)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(320, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 2, 3]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("320.33");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 320 = 2^{6} \cdot 5 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 320.m (of order \(4\), degree \(2\), 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: \(8.71936845953\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: 8.0.31116960000.4
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 7x^{6} + 16x^{4} + 105x^{2} + 225 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 2^{6}\cdot 5 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{4} + \beta_1 - 1) q^{3} + \beta_{3} q^{5} + ( - \beta_{6} - \beta_{5} - \beta_{3}) q^{7} + (\beta_{4} - \beta_{2} - 3 \beta_1 + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{4} + \beta_1 - 1) q^{3} + \beta_{3} q^{5} + ( - \beta_{6} - \beta_{5} - \beta_{3}) q^{7} + (\beta_{4} - \beta_{2} - 3 \beta_1 + 1) q^{9} + (\beta_{4} - \beta_{2} - 4 \beta_1 + 1) q^{11} + (\beta_{7} - \beta_{6} + \cdots + \beta_{3}) q^{13}+ \cdots + ( - 5 \beta_{4} - 5 \beta_{2} - 27) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 4 q^{3}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 4 q^{3} - 56 q^{17} + 16 q^{19} + 56 q^{27} + 96 q^{33} - 180 q^{35} + 104 q^{41} + 140 q^{43} - 344 q^{57} - 224 q^{59} - 240 q^{65} + 444 q^{67} + 336 q^{73} - 380 q^{75} + 592 q^{81} + 244 q^{83} - 720 q^{97} - 216 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 7x^{6} + 16x^{4} + 105x^{2} + 225 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 11\nu^{7} - 92\nu^{5} + 56\nu^{3} + 1815\nu ) / 2700 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -13\nu^{7} - 15\nu^{6} + 76\nu^{5} + 60\nu^{4} - 148\nu^{3} + 300\nu^{2} - 2145\nu - 1395 ) / 1080 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -13\nu^{7} - 51\nu^{6} + 76\nu^{5} + 492\nu^{4} - 148\nu^{3} - 1716\nu^{2} + 15\nu - 3015 ) / 1080 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 13\nu^{7} - 15\nu^{6} - 76\nu^{5} + 60\nu^{4} + 148\nu^{3} + 300\nu^{2} + 2145\nu - 2475 ) / 1080 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -13\nu^{7} + 99\nu^{6} + 76\nu^{5} - 828\nu^{4} - 148\nu^{3} + 3204\nu^{2} + 15\nu + 5535 ) / 1080 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 31\nu^{7} - 3\nu^{6} - 292\nu^{5} + 156\nu^{4} + 1156\nu^{3} - 228\nu^{2} + 795\nu - 495 ) / 1080 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -49\nu^{7} - 3\nu^{6} + 508\nu^{5} + 156\nu^{4} - 2164\nu^{3} - 228\nu^{2} - 1605\nu - 495 ) / 1080 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( -\beta_{7} - 2\beta_{6} + 3\beta_{5} + 5\beta_{4} + 6\beta_{3} - 5\beta_{2} + 5 ) / 20 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{7} + 2\beta_{6} + \beta_{5} + 3\beta_{4} + 3\beta_{2} + 7 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -\beta_{7} + \beta_{5} + 3\beta_{4} + 2\beta_{3} - 3\beta_{2} - 20\beta _1 + 3 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 11\beta_{7} + 22\beta_{6} + 5\beta_{5} + 3\beta_{4} + 6\beta_{3} + 3\beta_{2} + 17 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 5\beta_{7} - 5\beta_{5} + 51\beta_{4} - 10\beta_{3} - 51\beta_{2} - 290\beta _1 + 51 ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 16\beta_{7} + 32\beta_{6} + 10\beta_{5} - 18\beta_{4} + 6\beta_{3} - 18\beta_{2} - 77 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 85\beta_{7} + 66\beta_{6} - 151\beta_{5} + 231\beta_{4} - 302\beta_{3} - 231\beta_{2} - 1240\beta _1 + 231 ) / 4 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/320\mathbb{Z}\right)^\times\).

\(n\) \(191\) \(257\) \(261\)
\(\chi(n)\) \(1\) \(-\beta_{1}\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
33.1
−2.44727 − 0.895644i
2.44727 − 0.895644i
−0.510774 + 1.39564i
0.510774 + 1.39564i
−2.44727 + 0.895644i
2.44727 + 0.895644i
−0.510774 − 1.39564i
0.510774 − 1.39564i
0 −2.79129 + 2.79129i 0 −4.89453 + 1.02155i 0 8.76751 − 8.76751i 0 − 6.58258i 0
33.2 0 −2.79129 + 2.79129i 0 4.89453 − 1.02155i 0 −8.76751 + 8.76751i 0 − 6.58258i 0
33.3 0 1.79129 − 1.79129i 0 −1.02155 + 4.89453i 0 −2.85144 + 2.85144i 0 2.58258i 0
33.4 0 1.79129 − 1.79129i 0 1.02155 − 4.89453i 0 2.85144 − 2.85144i 0 2.58258i 0
97.1 0 −2.79129 − 2.79129i 0 −4.89453 − 1.02155i 0 8.76751 + 8.76751i 0 6.58258i 0
97.2 0 −2.79129 − 2.79129i 0 4.89453 + 1.02155i 0 −8.76751 − 8.76751i 0 6.58258i 0
97.3 0 1.79129 + 1.79129i 0 −1.02155 − 4.89453i 0 −2.85144 − 2.85144i 0 − 2.58258i 0
97.4 0 1.79129 + 1.79129i 0 1.02155 + 4.89453i 0 2.85144 + 2.85144i 0 − 2.58258i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 33.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 inner
20.e even 4 1 inner
40.i odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 320.3.m.a ✓ 8
4.b odd 2 1 320.3.m.b yes 8
5.c odd 4 1 320.3.m.b yes 8
8.b even 2 1 320.3.m.b yes 8
8.d odd 2 1 inner 320.3.m.a ✓ 8
20.e even 4 1 inner 320.3.m.a ✓ 8
40.i odd 4 1 inner 320.3.m.a ✓ 8
40.k even 4 1 320.3.m.b yes 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
320.3.m.a ✓ 8 1.a even 1 1 trivial
320.3.m.a ✓ 8 8.d odd 2 1 inner
320.3.m.a ✓ 8 20.e even 4 1 inner
320.3.m.a ✓ 8 40.i odd 4 1 inner
320.3.m.b yes 8 4.b odd 2 1
320.3.m.b yes 8 5.c odd 4 1
320.3.m.b yes 8 8.b even 2 1
320.3.m.b yes 8 40.k even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} + 2T_{3}^{3} + 2T_{3}^{2} - 20T_{3} + 100 \) acting on \(S_{3}^{\mathrm{new}}(320, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( (T^{4} + 2 T^{3} + \cdots + 100)^{2} \) Copy content Toggle raw display
$5$ \( T^{8} - 850 T^{4} + 390625 \) Copy content Toggle raw display
$7$ \( T^{8} + 23900 T^{4} + 6250000 \) Copy content Toggle raw display
$11$ \( (T^{4} + 60 T^{2} + 144)^{2} \) Copy content Toggle raw display
$13$ \( T^{8} + 139400 T^{4} + 6250000 \) Copy content Toggle raw display
$17$ \( (T^{4} + 28 T^{3} + \cdots + 4900)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} - 4 T - 332)^{4} \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 522006250000 \) Copy content Toggle raw display
$29$ \( (T^{2} - 1500)^{4} \) Copy content Toggle raw display
$31$ \( (T^{4} - 2500 T^{2} + 250000)^{2} \) Copy content Toggle raw display
$37$ \( T^{8} + 4883400 T^{4} + 506250000 \) Copy content Toggle raw display
$41$ \( (T^{2} - 26 T - 1532)^{4} \) Copy content Toggle raw display
$43$ \( (T^{4} - 70 T^{3} + \cdots + 122500)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 522006250000 \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots + 3906250000 \) Copy content Toggle raw display
$59$ \( (T^{2} + 56 T - 1316)^{4} \) Copy content Toggle raw display
$61$ \( (T^{4} + 8500 T^{2} + 6250000)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} - 222 T^{3} + \cdots + 28196100)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} - 20500 T^{2} + 72250000)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} - 168 T^{3} + \cdots + 2160900)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} + 14000)^{4} \) Copy content Toggle raw display
$83$ \( (T^{4} - 122 T^{3} + \cdots + 3422500)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} + 5040 T^{2} + 1016064)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} + 360 T^{3} + \cdots + 229522500)^{2} \) Copy content Toggle raw display
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