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Properties

Label 3920.1.fc.a
Level $3920$
Weight $1$
Character orbit 3920.fc
Analytic conductor $1.956$
Analytic rank $0$
Dimension $24$
Projective image $D_{42}$
CM discriminant -20
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3920,1,Mod(319,3920)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3920, base_ring=CyclotomicField(42))
 
chi = DirichletCharacter(H, H._module([21, 0, 21, 16]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3920.319");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3920 = 2^{4} \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3920.fc (of order \(42\), degree \(12\), 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: \(1.95633484952\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(2\) over \(\Q(\zeta_{42})\)
Coefficient field: \(\Q(\zeta_{84})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{24} + x^{22} - x^{18} - x^{16} + x^{12} - x^{8} - x^{6} + x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{42}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{42} - \cdots)\)

$q$-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.

\(f(q)\) \(=\) \( q + (\zeta_{84}^{35} - \zeta_{84}^{15}) q^{3} - \zeta_{84}^{32} q^{5} + \zeta_{84}^{19} q^{7} + (\zeta_{84}^{30} + \cdots + \zeta_{84}^{8}) q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + (\zeta_{84}^{35} - \zeta_{84}^{15}) q^{3} - \zeta_{84}^{32} q^{5} + \zeta_{84}^{19} q^{7} + (\zeta_{84}^{30} + \cdots + \zeta_{84}^{8}) q^{9} + \cdots + ( - \zeta_{84}^{34} + \zeta_{84}^{24}) q^{89} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 24 q - 2 q^{5} + 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 24 q - 2 q^{5} + 18 q^{9} + 6 q^{21} + 2 q^{25} - 10 q^{29} - 4 q^{41} - 4 q^{45} - 2 q^{49} + 2 q^{61} - 12 q^{69} - 12 q^{81} - 2 q^{89}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3920\mathbb{Z}\right)^\times\).

\(n\) \(981\) \(1471\) \(3041\) \(3137\)
\(\chi(n)\) \(1\) \(-1\) \(\zeta_{84}^{8}\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
319.1
0.294755 − 0.955573i
−0.294755 + 0.955573i
0.294755 + 0.955573i
−0.294755 − 0.955573i
−0.930874 + 0.365341i
0.930874 − 0.365341i
0.997204 − 0.0747301i
−0.997204 + 0.0747301i
−0.149042 + 0.988831i
0.149042 − 0.988831i
−0.149042 − 0.988831i
0.149042 + 0.988831i
−0.930874 − 0.365341i
0.930874 + 0.365341i
0.680173 − 0.733052i
−0.680173 + 0.733052i
0.563320 − 0.826239i
−0.563320 + 0.826239i
0.997204 + 0.0747301i
−0.997204 − 0.0747301i
0 −0.108903 − 0.277479i 0 0.988831 + 0.149042i 0 0.563320 + 0.826239i 0 0.667917 − 0.619736i 0
319.2 0 0.108903 + 0.277479i 0 0.988831 + 0.149042i 0 −0.563320 − 0.826239i 0 0.667917 − 0.619736i 0
639.1 0 −0.108903 + 0.277479i 0 0.988831 − 0.149042i 0 0.563320 − 0.826239i 0 0.667917 + 0.619736i 0
639.2 0 0.108903 − 0.277479i 0 0.988831 − 0.149042i 0 −0.563320 + 0.826239i 0 0.667917 + 0.619736i 0
879.1 0 −0.0841939 + 1.12349i 0 −0.826239 − 0.563320i 0 −0.680173 + 0.733052i 0 −0.266310 − 0.0401398i 0
879.2 0 0.0841939 − 1.12349i 0 −0.826239 − 0.563320i 0 0.680173 − 0.733052i 0 −0.266310 − 0.0401398i 0
1199.1 0 −1.29991 + 0.400969i 0 0.733052 + 0.680173i 0 0.149042 − 0.988831i 0 0.702749 − 0.479126i 0
1199.2 0 1.29991 − 0.400969i 0 0.733052 + 0.680173i 0 −0.149042 + 0.988831i 0 0.702749 − 0.479126i 0
1759.1 0 −1.64786 − 1.12349i 0 −0.0747301 + 0.997204i 0 0.294755 + 0.955573i 0 1.08786 + 2.77183i 0
1759.2 0 1.64786 + 1.12349i 0 −0.0747301 + 0.997204i 0 −0.294755 − 0.955573i 0 1.08786 + 2.77183i 0
1999.1 0 −1.64786 + 1.12349i 0 −0.0747301 − 0.997204i 0 0.294755 − 0.955573i 0 1.08786 − 2.77183i 0
1999.2 0 1.64786 − 1.12349i 0 −0.0747301 − 0.997204i 0 −0.294755 + 0.955573i 0 1.08786 − 2.77183i 0
2319.1 0 −0.0841939 − 1.12349i 0 −0.826239 + 0.563320i 0 −0.680173 − 0.733052i 0 −0.266310 + 0.0401398i 0
2319.2 0 0.0841939 + 1.12349i 0 −0.826239 + 0.563320i 0 0.680173 + 0.733052i 0 −0.266310 + 0.0401398i 0
2559.1 0 −1.84095 + 0.277479i 0 −0.365341 + 0.930874i 0 −0.997204 − 0.0747301i 0 2.35654 − 0.726897i 0
2559.2 0 1.84095 − 0.277479i 0 −0.365341 + 0.930874i 0 0.997204 + 0.0747301i 0 2.35654 − 0.726897i 0
2879.1 0 −0.432142 + 0.400969i 0 −0.955573 − 0.294755i 0 0.930874 + 0.365341i 0 −0.0487597 + 0.650653i 0
2879.2 0 0.432142 − 0.400969i 0 −0.955573 − 0.294755i 0 −0.930874 − 0.365341i 0 −0.0487597 + 0.650653i 0
3119.1 0 −1.29991 − 0.400969i 0 0.733052 − 0.680173i 0 0.149042 + 0.988831i 0 0.702749 + 0.479126i 0
3119.2 0 1.29991 + 0.400969i 0 0.733052 − 0.680173i 0 −0.149042 − 0.988831i 0 0.702749 + 0.479126i 0
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 319.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
20.d odd 2 1 CM by \(\Q(\sqrt{-5}) \)
4.b odd 2 1 inner
5.b even 2 1 inner
49.g even 21 1 inner
196.o odd 42 1 inner
245.t even 42 1 inner
980.bq odd 42 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3920.1.fc.a ✓ 24
4.b odd 2 1 inner 3920.1.fc.a ✓ 24
5.b even 2 1 inner 3920.1.fc.a ✓ 24
20.d odd 2 1 CM 3920.1.fc.a ✓ 24
49.g even 21 1 inner 3920.1.fc.a ✓ 24
196.o odd 42 1 inner 3920.1.fc.a ✓ 24
245.t even 42 1 inner 3920.1.fc.a ✓ 24
980.bq odd 42 1 inner 3920.1.fc.a ✓ 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3920.1.fc.a ✓ 24 1.a even 1 1 trivial
3920.1.fc.a ✓ 24 4.b odd 2 1 inner
3920.1.fc.a ✓ 24 5.b even 2 1 inner
3920.1.fc.a ✓ 24 20.d odd 2 1 CM
3920.1.fc.a ✓ 24 49.g even 21 1 inner
3920.1.fc.a ✓ 24 196.o odd 42 1 inner
3920.1.fc.a ✓ 24 245.t even 42 1 inner
3920.1.fc.a ✓ 24 980.bq odd 42 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(3920, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{24} \) Copy content Toggle raw display
$3$ \( T^{24} - 10 T^{22} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( (T^{12} + T^{11} - T^{9} + \cdots + 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{24} + T^{22} + \cdots + 1 \) Copy content Toggle raw display
$11$ \( T^{24} \) Copy content Toggle raw display
$13$ \( T^{24} \) Copy content Toggle raw display
$17$ \( T^{24} \) Copy content Toggle raw display
$19$ \( T^{24} \) Copy content Toggle raw display
$23$ \( T^{24} - 3 T^{22} + \cdots + 1 \) Copy content Toggle raw display
$29$ \( (T^{12} + 5 T^{11} + \cdots + 1)^{2} \) Copy content Toggle raw display
$31$ \( T^{24} \) Copy content Toggle raw display
$37$ \( T^{24} \) Copy content Toggle raw display
$41$ \( (T^{12} + 2 T^{11} + \cdots + 1)^{2} \) Copy content Toggle raw display
$43$ \( T^{24} + 6 T^{22} + \cdots + 1 \) Copy content Toggle raw display
$47$ \( T^{24} + 70 T^{18} + \cdots + 2401 \) Copy content Toggle raw display
$53$ \( T^{24} \) Copy content Toggle raw display
$59$ \( T^{24} \) Copy content Toggle raw display
$61$ \( (T^{12} - T^{11} + T^{9} + \cdots + 1)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} + 3 T^{2} + 9)^{6} \) Copy content Toggle raw display
$71$ \( T^{24} \) Copy content Toggle raw display
$73$ \( T^{24} \) Copy content Toggle raw display
$79$ \( T^{24} \) Copy content Toggle raw display
$83$ \( T^{24} + 6 T^{22} + \cdots + 1 \) Copy content Toggle raw display
$89$ \( (T^{12} + T^{11} - T^{9} + \cdots + 1)^{2} \) Copy content Toggle raw display
$97$ \( T^{24} \) Copy content Toggle raw display
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