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Properties

Label 3744.1.eh.a
Level $3744$
Weight $1$
Character orbit 3744.eh
Analytic conductor $1.868$
Analytic rank $0$
Dimension $16$
Projective image $D_{16}$
CM discriminant -39
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3744,1,Mod(883,3744)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3744, base_ring=CyclotomicField(8))
 
chi = DirichletCharacter(H, H._module([4, 7, 0, 4]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3744.883");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3744 = 2^{5} \cdot 3^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3744.eh (of order \(8\), degree \(4\), 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: \(1.86849940730\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(4\) over \(\Q(\zeta_{8})\)
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: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q - \zeta_{32}^{3} q^{2} + \zeta_{32}^{6} q^{4} + ( - \zeta_{32}^{11} - \zeta_{32}) q^{5} - \zeta_{32}^{9} q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q - \zeta_{32}^{3} q^{2} + \zeta_{32}^{6} q^{4} + ( - \zeta_{32}^{11} - \zeta_{32}) q^{5} - \zeta_{32}^{9} q^{8} + (\zeta_{32}^{14} + \zeta_{32}^{4}) q^{10} + ( - \zeta_{32}^{15} + \zeta_{32}^{13}) q^{11} + \zeta_{32}^{2} q^{13} + \zeta_{32}^{12} q^{16} + ( - \zeta_{32}^{7} + \zeta_{32}) q^{20} + ( - \zeta_{32}^{2} + 1) q^{22} + (\zeta_{32}^{12} + \cdots + \zeta_{32}^{2}) q^{25} + \cdots - \zeta_{32}^{11} q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 16 q^{22} - 16 q^{55}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3744\mathbb{Z}\right)^\times\).

\(n\) \(703\) \(2017\) \(2081\) \(2341\)
\(\chi(n)\) \(-1\) \(-1\) \(1\) \(-\zeta_{32}^{12}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
883.1
0.980785 + 0.195090i
−0.195090 + 0.980785i
0.195090 − 0.980785i
−0.980785 − 0.195090i
0.980785 − 0.195090i
−0.195090 − 0.980785i
0.195090 + 0.980785i
−0.980785 + 0.195090i
−0.555570 − 0.831470i
−0.831470 + 0.555570i
0.831470 − 0.555570i
0.555570 + 0.831470i
−0.555570 + 0.831470i
−0.831470 − 0.555570i
0.831470 + 0.555570i
0.555570 − 0.831470i
−0.831470 − 0.555570i 0 0.382683 + 0.923880i −0.425215 − 1.02656i 0 0 0.195090 − 0.980785i 0 −0.216773 + 1.08979i
883.2 −0.555570 + 0.831470i 0 −0.382683 − 0.923880i −0.636379 − 1.53636i 0 0 0.980785 + 0.195090i 0 1.63099 + 0.324423i
883.3 0.555570 − 0.831470i 0 −0.382683 − 0.923880i 0.636379 + 1.53636i 0 0 −0.980785 − 0.195090i 0 1.63099 + 0.324423i
883.4 0.831470 + 0.555570i 0 0.382683 + 0.923880i 0.425215 + 1.02656i 0 0 −0.195090 + 0.980785i 0 −0.216773 + 1.08979i
1819.1 −0.831470 + 0.555570i 0 0.382683 − 0.923880i −0.425215 + 1.02656i 0 0 0.195090 + 0.980785i 0 −0.216773 − 1.08979i
1819.2 −0.555570 − 0.831470i 0 −0.382683 + 0.923880i −0.636379 + 1.53636i 0 0 0.980785 − 0.195090i 0 1.63099 − 0.324423i
1819.3 0.555570 + 0.831470i 0 −0.382683 + 0.923880i 0.636379 − 1.53636i 0 0 −0.980785 + 0.195090i 0 1.63099 − 0.324423i
1819.4 0.831470 − 0.555570i 0 0.382683 − 0.923880i 0.425215 − 1.02656i 0 0 −0.195090 − 0.980785i 0 −0.216773 − 1.08979i
2755.1 −0.980785 + 0.195090i 0 0.923880 − 0.382683i 0.360480 − 0.149316i 0 0 −0.831470 + 0.555570i 0 −0.324423 + 0.216773i
2755.2 −0.195090 − 0.980785i 0 −0.923880 + 0.382683i 1.81225 − 0.750661i 0 0 0.555570 + 0.831470i 0 −1.08979 − 1.63099i
2755.3 0.195090 + 0.980785i 0 −0.923880 + 0.382683i −1.81225 + 0.750661i 0 0 −0.555570 − 0.831470i 0 −1.08979 − 1.63099i
2755.4 0.980785 − 0.195090i 0 0.923880 − 0.382683i −0.360480 + 0.149316i 0 0 0.831470 − 0.555570i 0 −0.324423 + 0.216773i
3691.1 −0.980785 − 0.195090i 0 0.923880 + 0.382683i 0.360480 + 0.149316i 0 0 −0.831470 − 0.555570i 0 −0.324423 − 0.216773i
3691.2 −0.195090 + 0.980785i 0 −0.923880 − 0.382683i 1.81225 + 0.750661i 0 0 0.555570 − 0.831470i 0 −1.08979 + 1.63099i
3691.3 0.195090 − 0.980785i 0 −0.923880 − 0.382683i −1.81225 − 0.750661i 0 0 −0.555570 + 0.831470i 0 −1.08979 + 1.63099i
3691.4 0.980785 + 0.195090i 0 0.923880 + 0.382683i −0.360480 − 0.149316i 0 0 0.831470 + 0.555570i 0 −0.324423 − 0.216773i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 883.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
39.d odd 2 1 CM by \(\Q(\sqrt{-39}) \)
3.b odd 2 1 inner
13.b even 2 1 inner
32.h odd 8 1 inner
96.o even 8 1 inner
416.be odd 8 1 inner
1248.cp even 8 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3744.1.eh.a ✓ 16
3.b odd 2 1 inner 3744.1.eh.a ✓ 16
13.b even 2 1 inner 3744.1.eh.a ✓ 16
32.h odd 8 1 inner 3744.1.eh.a ✓ 16
39.d odd 2 1 CM 3744.1.eh.a ✓ 16
96.o even 8 1 inner 3744.1.eh.a ✓ 16
416.be odd 8 1 inner 3744.1.eh.a ✓ 16
1248.cp even 8 1 inner 3744.1.eh.a ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3744.1.eh.a ✓ 16 1.a even 1 1 trivial
3744.1.eh.a ✓ 16 3.b odd 2 1 inner
3744.1.eh.a ✓ 16 13.b even 2 1 inner
3744.1.eh.a ✓ 16 32.h odd 8 1 inner
3744.1.eh.a ✓ 16 39.d odd 2 1 CM
3744.1.eh.a ✓ 16 96.o even 8 1 inner
3744.1.eh.a ✓ 16 416.be odd 8 1 inner
3744.1.eh.a ✓ 16 1248.cp even 8 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(3744, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} + 1 \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( T^{16} + 16 T^{10} + \cdots + 4 \) Copy content Toggle raw display
$7$ \( T^{16} \) Copy content Toggle raw display
$11$ \( T^{16} - 16 T^{10} + \cdots + 4 \) Copy content Toggle raw display
$13$ \( (T^{8} + 1)^{2} \) Copy content Toggle raw display
$17$ \( T^{16} \) Copy content Toggle raw display
$19$ \( T^{16} \) Copy content Toggle raw display
$23$ \( T^{16} \) Copy content Toggle raw display
$29$ \( T^{16} \) Copy content Toggle raw display
$31$ \( T^{16} \) Copy content Toggle raw display
$37$ \( T^{16} \) Copy content Toggle raw display
$41$ \( T^{16} + 24 T^{12} + \cdots + 4 \) Copy content Toggle raw display
$43$ \( (T^{4} + 2 T^{2} - 4 T + 2)^{4} \) Copy content Toggle raw display
$47$ \( (T^{8} + 8 T^{6} + 20 T^{4} + \cdots + 2)^{2} \) Copy content Toggle raw display
$53$ \( T^{16} \) Copy content Toggle raw display
$59$ \( T^{16} - 16 T^{10} + \cdots + 4 \) Copy content Toggle raw display
$61$ \( (T^{8} + 16)^{2} \) Copy content Toggle raw display
$67$ \( T^{16} \) Copy content Toggle raw display
$71$ \( T^{16} + 24 T^{12} + \cdots + 4 \) Copy content Toggle raw display
$73$ \( T^{16} \) Copy content Toggle raw display
$79$ \( (T^{4} - 4 T^{2} + 2)^{4} \) Copy content Toggle raw display
$83$ \( T^{16} - 16 T^{10} + \cdots + 4 \) Copy content Toggle raw display
$89$ \( T^{16} + 24 T^{12} + \cdots + 4 \) Copy content Toggle raw display
$97$ \( T^{16} \) Copy content Toggle raw display
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