Properties

Label 1920.1.bu.a
Level $1920$
Weight $1$
Character orbit 1920.bu
Analytic conductor $0.958$
Analytic rank $0$
Dimension $8$
Projective image $D_{8}$
CM discriminant -15
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1920,1,Mod(209,1920)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1920, base_ring=CyclotomicField(8))
 
chi = DirichletCharacter(H, H._module([0, 3, 4, 4]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1920.209");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1920 = 2^{7} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1920.bu (of order \(8\), degree \(4\), 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: \(0.958204824255\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{8})\)
Coefficient field: \(\Q(\zeta_{16})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 480)
Projective image: \(D_{8}\)
Projective field: Galois closure of 8.2.36238786560000.10

$q$-expansion

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

The \(q\)-expansion and trace form are shown below.

\(f(q)\) \(=\) \( q + \zeta_{16}^{5} q^{3} - \zeta_{16}^{7} q^{5} - \zeta_{16}^{2} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{16}^{5} q^{3} - \zeta_{16}^{7} q^{5} - \zeta_{16}^{2} q^{9} + \zeta_{16}^{4} q^{15} + (\zeta_{16}^{5} + \zeta_{16}^{3}) q^{17} + (\zeta_{16}^{6} - \zeta_{16}^{4}) q^{19} + (\zeta_{16}^{3} + \zeta_{16}) q^{23} - \zeta_{16}^{6} q^{25} - \zeta_{16}^{7} q^{27} + ( - \zeta_{16}^{6} + \zeta_{16}^{2}) q^{31} - \zeta_{16} q^{45} + ( - \zeta_{16}^{7} - \zeta_{16}) q^{47} - \zeta_{16}^{4} q^{49} + ( - \zeta_{16}^{2} - 1) q^{51} + (\zeta_{16}^{5} + \zeta_{16}) q^{53} + ( - \zeta_{16}^{3} + \zeta_{16}) q^{57} + (\zeta_{16}^{2} - 1) q^{61} + (\zeta_{16}^{6} - 1) q^{69} + \zeta_{16}^{3} q^{75} + \zeta_{16}^{4} q^{81} + (\zeta_{16}^{7} - \zeta_{16}^{3}) q^{83} + (\zeta_{16}^{4} + \zeta_{16}^{2}) q^{85} + (\zeta_{16}^{7} + \zeta_{16}^{3}) q^{93} + (\zeta_{16}^{5} - \zeta_{16}^{3}) q^{95} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 8 q^{51} - 8 q^{61} - 8 q^{69}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1920\mathbb{Z}\right)^\times\).

\(n\) \(511\) \(641\) \(901\) \(1537\)
\(\chi(n)\) \(1\) \(-1\) \(-\zeta_{16}^{6}\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
209.1
−0.382683 + 0.923880i
0.382683 0.923880i
−0.382683 0.923880i
0.382683 + 0.923880i
0.923880 + 0.382683i
−0.923880 0.382683i
0.923880 0.382683i
−0.923880 + 0.382683i
0 −0.923880 0.382683i 0 −0.382683 0.923880i 0 0 0 0.707107 + 0.707107i 0
209.2 0 0.923880 + 0.382683i 0 0.382683 + 0.923880i 0 0 0 0.707107 + 0.707107i 0
689.1 0 −0.923880 + 0.382683i 0 −0.382683 + 0.923880i 0 0 0 0.707107 0.707107i 0
689.2 0 0.923880 0.382683i 0 0.382683 0.923880i 0 0 0 0.707107 0.707107i 0
1169.1 0 −0.382683 + 0.923880i 0 0.923880 0.382683i 0 0 0 −0.707107 0.707107i 0
1169.2 0 0.382683 0.923880i 0 −0.923880 + 0.382683i 0 0 0 −0.707107 0.707107i 0
1649.1 0 −0.382683 0.923880i 0 0.923880 + 0.382683i 0 0 0 −0.707107 + 0.707107i 0
1649.2 0 0.382683 + 0.923880i 0 −0.923880 0.382683i 0 0 0 −0.707107 + 0.707107i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 209.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
15.d odd 2 1 CM by \(\Q(\sqrt{-15}) \)
3.b odd 2 1 inner
5.b even 2 1 inner
32.g even 8 1 inner
96.p odd 8 1 inner
160.z even 8 1 inner
480.bu odd 8 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1920.1.bu.a 8
3.b odd 2 1 inner 1920.1.bu.a 8
4.b odd 2 1 480.1.bu.a 8
5.b even 2 1 inner 1920.1.bu.a 8
8.b even 2 1 3840.1.bu.b 8
8.d odd 2 1 3840.1.bu.a 8
12.b even 2 1 480.1.bu.a 8
15.d odd 2 1 CM 1920.1.bu.a 8
20.d odd 2 1 480.1.bu.a 8
20.e even 4 2 2400.1.ca.a 8
24.f even 2 1 3840.1.bu.a 8
24.h odd 2 1 3840.1.bu.b 8
32.g even 8 1 inner 1920.1.bu.a 8
32.g even 8 1 3840.1.bu.b 8
32.h odd 8 1 480.1.bu.a 8
32.h odd 8 1 3840.1.bu.a 8
40.e odd 2 1 3840.1.bu.a 8
40.f even 2 1 3840.1.bu.b 8
60.h even 2 1 480.1.bu.a 8
60.l odd 4 2 2400.1.ca.a 8
96.o even 8 1 480.1.bu.a 8
96.o even 8 1 3840.1.bu.a 8
96.p odd 8 1 inner 1920.1.bu.a 8
96.p odd 8 1 3840.1.bu.b 8
120.i odd 2 1 3840.1.bu.b 8
120.m even 2 1 3840.1.bu.a 8
160.u even 8 1 2400.1.ca.a 8
160.y odd 8 1 480.1.bu.a 8
160.y odd 8 1 3840.1.bu.a 8
160.z even 8 1 inner 1920.1.bu.a 8
160.z even 8 1 3840.1.bu.b 8
160.ba even 8 1 2400.1.ca.a 8
480.bq odd 8 1 2400.1.ca.a 8
480.bs even 8 1 480.1.bu.a 8
480.bs even 8 1 3840.1.bu.a 8
480.bu odd 8 1 inner 1920.1.bu.a 8
480.bu odd 8 1 3840.1.bu.b 8
480.ca odd 8 1 2400.1.ca.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
480.1.bu.a 8 4.b odd 2 1
480.1.bu.a 8 12.b even 2 1
480.1.bu.a 8 20.d odd 2 1
480.1.bu.a 8 32.h odd 8 1
480.1.bu.a 8 60.h even 2 1
480.1.bu.a 8 96.o even 8 1
480.1.bu.a 8 160.y odd 8 1
480.1.bu.a 8 480.bs even 8 1
1920.1.bu.a 8 1.a even 1 1 trivial
1920.1.bu.a 8 3.b odd 2 1 inner
1920.1.bu.a 8 5.b even 2 1 inner
1920.1.bu.a 8 15.d odd 2 1 CM
1920.1.bu.a 8 32.g even 8 1 inner
1920.1.bu.a 8 96.p odd 8 1 inner
1920.1.bu.a 8 160.z even 8 1 inner
1920.1.bu.a 8 480.bu odd 8 1 inner
2400.1.ca.a 8 20.e even 4 2
2400.1.ca.a 8 60.l odd 4 2
2400.1.ca.a 8 160.u even 8 1
2400.1.ca.a 8 160.ba even 8 1
2400.1.ca.a 8 480.bq odd 8 1
2400.1.ca.a 8 480.ca odd 8 1
3840.1.bu.a 8 8.d odd 2 1
3840.1.bu.a 8 24.f even 2 1
3840.1.bu.a 8 32.h odd 8 1
3840.1.bu.a 8 40.e odd 2 1
3840.1.bu.a 8 96.o even 8 1
3840.1.bu.a 8 120.m even 2 1
3840.1.bu.a 8 160.y odd 8 1
3840.1.bu.a 8 480.bs even 8 1
3840.1.bu.b 8 8.b even 2 1
3840.1.bu.b 8 24.h odd 2 1
3840.1.bu.b 8 32.g even 8 1
3840.1.bu.b 8 40.f even 2 1
3840.1.bu.b 8 96.p odd 8 1
3840.1.bu.b 8 120.i odd 2 1
3840.1.bu.b 8 160.z even 8 1
3840.1.bu.b 8 480.bu odd 8 1

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(1920, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} + 1 \) Copy content Toggle raw display
$5$ \( T^{8} + 1 \) Copy content Toggle raw display
$7$ \( T^{8} \) Copy content Toggle raw display
$11$ \( T^{8} \) Copy content Toggle raw display
$13$ \( T^{8} \) Copy content Toggle raw display
$17$ \( (T^{4} + 4 T^{2} + 2)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} + 2 T^{2} + 4 T + 2)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} + 12T^{4} + 4 \) Copy content Toggle raw display
$29$ \( T^{8} \) Copy content Toggle raw display
$31$ \( (T^{2} - 2)^{4} \) Copy content Toggle raw display
$37$ \( T^{8} \) Copy content Toggle raw display
$41$ \( T^{8} \) Copy content Toggle raw display
$43$ \( T^{8} \) Copy content Toggle raw display
$47$ \( (T^{4} + 4 T^{2} + 2)^{2} \) Copy content Toggle raw display
$53$ \( T^{8} + 16 \) Copy content Toggle raw display
$59$ \( T^{8} \) Copy content Toggle raw display
$61$ \( (T^{4} + 4 T^{3} + 6 T^{2} + \cdots + 2)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} \) Copy content Toggle raw display
$71$ \( T^{8} \) Copy content Toggle raw display
$73$ \( T^{8} \) Copy content Toggle raw display
$79$ \( T^{8} \) Copy content Toggle raw display
$83$ \( T^{8} + 16 \) Copy content Toggle raw display
$89$ \( T^{8} \) Copy content Toggle raw display
$97$ \( T^{8} \) Copy content Toggle raw display
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