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Properties

Label 3840.1.cl.a
Level $3840$
Weight $1$
Character orbit 3840.cl
Analytic conductor $1.916$
Analytic rank $0$
Dimension $16$
Projective image $D_{16}$
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] = [3840,1,Mod(209,3840)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3840, base_ring=CyclotomicField(16))
 
chi = DirichletCharacter(H, H._module([0, 11, 8, 8]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3840.209");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3840 = 2^{8} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3840.cl (of order \(16\), degree \(8\), 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: \(1.91640964851\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(2\) over \(\Q(\zeta_{16})\)
Coefficient field: \(\Q(\zeta_{32})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 1 \) 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 960)
Projective image: \(D_{16}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{16} - \cdots)\)

$q$-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.

\(f(q)\) \(=\) \( q - \zeta_{32}^{3} q^{3} + \zeta_{32} q^{5} + \zeta_{32}^{6} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - \zeta_{32}^{3} q^{3} + \zeta_{32} q^{5} + \zeta_{32}^{6} q^{9} - \zeta_{32}^{4} q^{15} + (\zeta_{32}^{13} + \zeta_{32}^{11}) q^{17} + (\zeta_{32}^{12} + \zeta_{32}^{2}) q^{19} + ( - \zeta_{32}^{7} + \zeta_{32}^{5}) q^{23} + \zeta_{32}^{2} q^{25} - \zeta_{32}^{9} q^{27} + (\zeta_{32}^{10} + \zeta_{32}^{6}) q^{31} + \zeta_{32}^{7} q^{45} + (\zeta_{32}^{15} + \zeta_{32}^{9}) q^{47} - \zeta_{32}^{4} q^{49} + ( - \zeta_{32}^{14} + 1) q^{51} + (\zeta_{32}^{15} - \zeta_{32}^{11}) q^{53} + ( - \zeta_{32}^{15} - \zeta_{32}^{5}) q^{57} + ( - \zeta_{32}^{14} + \zeta_{32}^{8}) q^{61} + (\zeta_{32}^{10} - \zeta_{32}^{8}) q^{69} - \zeta_{32}^{5} q^{75} + (\zeta_{32}^{8} + 1) q^{79} + \zeta_{32}^{12} q^{81} + ( - \zeta_{32}^{13} - \zeta_{32}) q^{83} + (\zeta_{32}^{14} + \zeta_{32}^{12}) q^{85} + ( - \zeta_{32}^{13} - \zeta_{32}^{9}) q^{93} + (\zeta_{32}^{13} + \zeta_{32}^{3}) q^{95} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 16 q^{51} + 16 q^{79}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3840\mathbb{Z}\right)^\times\).

\(n\) \(511\) \(1537\) \(2561\) \(2821\)
\(\chi(n)\) \(1\) \(-1\) \(-1\) \(\zeta_{32}^{2}\)

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
−0.555570 + 0.831470i
0.555570 − 0.831470i
−0.831470 + 0.555570i
0.831470 − 0.555570i
0.980785 − 0.195090i
−0.980785 + 0.195090i
0.980785 + 0.195090i
−0.980785 − 0.195090i
−0.831470 − 0.555570i
0.831470 + 0.555570i
−0.555570 − 0.831470i
0.555570 + 0.831470i
−0.195090 − 0.980785i
0.195090 + 0.980785i
−0.195090 + 0.980785i
0.195090 − 0.980785i
0 −0.980785 − 0.195090i 0 −0.555570 + 0.831470i 0 0 0 0.923880 + 0.382683i 0
209.2 0 0.980785 + 0.195090i 0 0.555570 − 0.831470i 0 0 0 0.923880 + 0.382683i 0
689.1 0 −0.195090 − 0.980785i 0 −0.831470 + 0.555570i 0 0 0 −0.923880 + 0.382683i 0
689.2 0 0.195090 + 0.980785i 0 0.831470 − 0.555570i 0 0 0 −0.923880 + 0.382683i 0
1169.1 0 −0.831470 + 0.555570i 0 0.980785 − 0.195090i 0 0 0 0.382683 − 0.923880i 0
1169.2 0 0.831470 − 0.555570i 0 −0.980785 + 0.195090i 0 0 0 0.382683 − 0.923880i 0
1649.1 0 −0.831470 − 0.555570i 0 0.980785 + 0.195090i 0 0 0 0.382683 + 0.923880i 0
1649.2 0 0.831470 + 0.555570i 0 −0.980785 − 0.195090i 0 0 0 0.382683 + 0.923880i 0
2129.1 0 −0.195090 + 0.980785i 0 −0.831470 − 0.555570i 0 0 0 −0.923880 − 0.382683i 0
2129.2 0 0.195090 − 0.980785i 0 0.831470 + 0.555570i 0 0 0 −0.923880 − 0.382683i 0
2609.1 0 −0.980785 + 0.195090i 0 −0.555570 − 0.831470i 0 0 0 0.923880 − 0.382683i 0
2609.2 0 0.980785 − 0.195090i 0 0.555570 + 0.831470i 0 0 0 0.923880 − 0.382683i 0
3089.1 0 −0.555570 − 0.831470i 0 −0.195090 − 0.980785i 0 0 0 −0.382683 + 0.923880i 0
3089.2 0 0.555570 + 0.831470i 0 0.195090 + 0.980785i 0 0 0 −0.382683 + 0.923880i 0
3569.1 0 −0.555570 + 0.831470i 0 −0.195090 + 0.980785i 0 0 0 −0.382683 − 0.923880i 0
3569.2 0 0.555570 − 0.831470i 0 0.195090 − 0.980785i 0 0 0 −0.382683 − 0.923880i 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
64.i even 16 1 inner
192.q odd 16 1 inner
320.bf even 16 1 inner
960.cl odd 16 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3840.1.cl.a 16
3.b odd 2 1 inner 3840.1.cl.a 16
4.b odd 2 1 960.1.cl.a ✓ 16
5.b even 2 1 inner 3840.1.cl.a 16
12.b even 2 1 960.1.cl.a ✓ 16
15.d odd 2 1 CM 3840.1.cl.a 16
20.d odd 2 1 960.1.cl.a ✓ 16
60.h even 2 1 960.1.cl.a ✓ 16
64.i even 16 1 inner 3840.1.cl.a 16
64.j odd 16 1 960.1.cl.a ✓ 16
192.q odd 16 1 inner 3840.1.cl.a 16
192.s even 16 1 960.1.cl.a ✓ 16
320.bf even 16 1 inner 3840.1.cl.a 16
320.bh odd 16 1 960.1.cl.a ✓ 16
960.cl odd 16 1 inner 3840.1.cl.a 16
960.cp even 16 1 960.1.cl.a ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
960.1.cl.a ✓ 16 4.b odd 2 1
960.1.cl.a ✓ 16 12.b even 2 1
960.1.cl.a ✓ 16 20.d odd 2 1
960.1.cl.a ✓ 16 60.h even 2 1
960.1.cl.a ✓ 16 64.j odd 16 1
960.1.cl.a ✓ 16 192.s even 16 1
960.1.cl.a ✓ 16 320.bh odd 16 1
960.1.cl.a ✓ 16 960.cp even 16 1
3840.1.cl.a 16 1.a even 1 1 trivial
3840.1.cl.a 16 3.b odd 2 1 inner
3840.1.cl.a 16 5.b even 2 1 inner
3840.1.cl.a 16 15.d odd 2 1 CM
3840.1.cl.a 16 64.i even 16 1 inner
3840.1.cl.a 16 192.q odd 16 1 inner
3840.1.cl.a 16 320.bf even 16 1 inner
3840.1.cl.a 16 960.cl odd 16 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(3840, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( T^{16} + 1 \) Copy content Toggle raw display
$5$ \( T^{16} + 1 \) Copy content Toggle raw display
$7$ \( T^{16} \) Copy content Toggle raw display
$11$ \( T^{16} \) Copy content Toggle raw display
$13$ \( T^{16} \) Copy content Toggle raw display
$17$ \( T^{16} + 24 T^{12} + \cdots + 4 \) Copy content Toggle raw display
$19$ \( (T^{8} + 8 T^{5} + 2 T^{4} + \cdots + 2)^{2} \) Copy content Toggle raw display
$23$ \( T^{16} + 16 T^{10} + \cdots + 4 \) Copy content Toggle raw display
$29$ \( T^{16} \) Copy content Toggle raw display
$31$ \( (T^{4} + 4 T^{2} + 2)^{4} \) Copy content Toggle raw display
$37$ \( T^{16} \) Copy content Toggle raw display
$41$ \( T^{16} \) Copy content Toggle raw display
$43$ \( T^{16} \) Copy content Toggle raw display
$47$ \( T^{16} + 24 T^{12} + \cdots + 4 \) Copy content Toggle raw display
$53$ \( T^{16} - 16 T^{12} + \cdots + 16 \) Copy content Toggle raw display
$59$ \( T^{16} \) Copy content Toggle raw display
$61$ \( (T^{8} + 4 T^{6} + 6 T^{4} + \cdots + 2)^{2} \) Copy content Toggle raw display
$67$ \( T^{16} \) Copy content Toggle raw display
$71$ \( T^{16} \) Copy content Toggle raw display
$73$ \( T^{16} \) Copy content Toggle raw display
$79$ \( (T^{2} - 2 T + 2)^{8} \) Copy content Toggle raw display
$83$ \( T^{16} + 16 T^{12} + \cdots + 16 \) Copy content Toggle raw display
$89$ \( T^{16} \) Copy content Toggle raw display
$97$ \( T^{16} \) Copy content Toggle raw display
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