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Properties

Label 1476.1.cc.b
Level $1476$
Weight $1$
Character orbit 1476.cc
Analytic conductor $0.737$
Analytic rank $0$
Dimension $16$
Projective image $D_{40}$
CM discriminant -4
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1476,1,Mod(35,1476)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1476, base_ring=CyclotomicField(40))
 
chi = DirichletCharacter(H, H._module([20, 20, 21]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1476.35");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1476 = 2^{2} \cdot 3^{2} \cdot 41 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1476.cc (of order \(40\), degree \(16\), 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: \(0.736619958646\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\Q(\zeta_{40})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - x^{12} + x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{40}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{40} + \cdots)\)

$q$-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.

\(f(q)\) \(=\) \( q - \zeta_{40}^{13} q^{2} - \zeta_{40}^{6} q^{4} + ( - \zeta_{40}^{14} - \zeta_{40}^{8}) q^{5} + \zeta_{40}^{19} q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q - \zeta_{40}^{13} q^{2} - \zeta_{40}^{6} q^{4} + ( - \zeta_{40}^{14} - \zeta_{40}^{8}) q^{5} + \zeta_{40}^{19} q^{8} + ( - \zeta_{40}^{7} - \zeta_{40}) q^{10} + ( - \zeta_{40}^{17} + \zeta_{40}^{4}) q^{13} + \zeta_{40}^{12} q^{16} + (\zeta_{40}^{17} - \zeta_{40}^{6}) q^{17} + (\zeta_{40}^{14} - 1) q^{20} + (\zeta_{40}^{16} + \cdots - \zeta_{40}^{2}) q^{25} + \cdots + \zeta_{40}^{2} q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 4 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 4 q^{5} + 4 q^{13} + 4 q^{16} - 16 q^{20} + 4 q^{29} + 4 q^{61} - 4 q^{65} + 4 q^{68} - 4 q^{74} + 16 q^{80} - 16 q^{85}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1476\mathbb{Z}\right)^\times\).

\(n\) \(739\) \(821\) \(1441\)
\(\chi(n)\) \(-1\) \(-1\) \(-\zeta_{40}^{11}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
35.1
−0.156434 + 0.987688i
0.453990 − 0.891007i
−0.453990 − 0.891007i
0.453990 + 0.891007i
−0.453990 + 0.891007i
0.156434 − 0.987688i
−0.987688 + 0.156434i
−0.891007 − 0.453990i
0.891007 − 0.453990i
0.156434 + 0.987688i
−0.987688 − 0.156434i
0.987688 + 0.156434i
−0.156434 − 0.987688i
−0.891007 + 0.453990i
0.891007 + 0.453990i
0.987688 − 0.156434i
0.891007 + 0.453990i 0 0.587785 + 0.809017i −0.896802 − 0.142040i 0 0 0.156434 + 0.987688i 0 −0.734572 − 0.533698i
71.1 0.156434 + 0.987688i 0 −0.951057 + 0.309017i 1.76007 + 0.896802i 0 0 −0.453990 − 0.891007i 0 −0.610425 + 1.87869i
179.1 −0.156434 + 0.987688i 0 −0.951057 − 0.309017i 1.76007 − 0.896802i 0 0 0.453990 − 0.891007i 0 0.610425 + 1.87869i
395.1 0.156434 − 0.987688i 0 −0.951057 − 0.309017i 1.76007 − 0.896802i 0 0 −0.453990 + 0.891007i 0 −0.610425 − 1.87869i
503.1 −0.156434 − 0.987688i 0 −0.951057 + 0.309017i 1.76007 + 0.896802i 0 0 0.453990 + 0.891007i 0 0.610425 − 1.87869i
539.1 −0.891007 − 0.453990i 0 0.587785 + 0.809017i −0.896802 − 0.142040i 0 0 −0.156434 − 0.987688i 0 0.734572 + 0.533698i
719.1 −0.453990 − 0.891007i 0 −0.587785 + 0.809017i 0.278768 + 1.76007i 0 0 0.987688 + 0.156434i 0 1.44168 − 1.04744i
755.1 0.987688 − 0.156434i 0 0.951057 − 0.309017i −0.142040 + 0.278768i 0 0 0.891007 − 0.453990i 0 −0.0966818 + 0.297556i
791.1 −0.987688 − 0.156434i 0 0.951057 + 0.309017i −0.142040 − 0.278768i 0 0 −0.891007 − 0.453990i 0 0.0966818 + 0.297556i
827.1 −0.891007 + 0.453990i 0 0.587785 − 0.809017i −0.896802 + 0.142040i 0 0 −0.156434 + 0.987688i 0 0.734572 − 0.533698i
971.1 −0.453990 + 0.891007i 0 −0.587785 − 0.809017i 0.278768 − 1.76007i 0 0 0.987688 − 0.156434i 0 1.44168 + 1.04744i
1079.1 0.453990 − 0.891007i 0 −0.587785 − 0.809017i 0.278768 − 1.76007i 0 0 −0.987688 + 0.156434i 0 −1.44168 − 1.04744i
1223.1 0.891007 − 0.453990i 0 0.587785 − 0.809017i −0.896802 + 0.142040i 0 0 0.156434 − 0.987688i 0 −0.734572 + 0.533698i
1259.1 0.987688 + 0.156434i 0 0.951057 + 0.309017i −0.142040 − 0.278768i 0 0 0.891007 + 0.453990i 0 −0.0966818 − 0.297556i
1295.1 −0.987688 + 0.156434i 0 0.951057 − 0.309017i −0.142040 + 0.278768i 0 0 −0.891007 + 0.453990i 0 0.0966818 − 0.297556i
1331.1 0.453990 + 0.891007i 0 −0.587785 + 0.809017i 0.278768 + 1.76007i 0 0 −0.987688 − 0.156434i 0 −1.44168 + 1.04744i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 35.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 CM by \(\Q(\sqrt{-1}) \)
123.o even 40 1 inner
492.be odd 40 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1476.1.cc.b yes 16
3.b odd 2 1 1476.1.cc.a ✓ 16
4.b odd 2 1 CM 1476.1.cc.b yes 16
12.b even 2 1 1476.1.cc.a ✓ 16
41.h odd 40 1 1476.1.cc.a ✓ 16
123.o even 40 1 inner 1476.1.cc.b yes 16
164.o even 40 1 1476.1.cc.a ✓ 16
492.be odd 40 1 inner 1476.1.cc.b yes 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1476.1.cc.a ✓ 16 3.b odd 2 1
1476.1.cc.a ✓ 16 12.b even 2 1
1476.1.cc.a ✓ 16 41.h odd 40 1
1476.1.cc.a ✓ 16 164.o even 40 1
1476.1.cc.b yes 16 1.a even 1 1 trivial
1476.1.cc.b yes 16 4.b odd 2 1 CM
1476.1.cc.b yes 16 123.o even 40 1 inner
1476.1.cc.b yes 16 492.be odd 40 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{8} - 2T_{5}^{7} + 2T_{5}^{6} - 4T_{5}^{4} + 10T_{5}^{3} + 13T_{5}^{2} + 4T_{5} + 1 \) acting on \(S_{1}^{\mathrm{new}}(1476, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} - T^{12} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( (T^{8} - 2 T^{7} + 2 T^{6} + \cdots + 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{16} \) Copy content Toggle raw display
$11$ \( T^{16} \) Copy content Toggle raw display
$13$ \( T^{16} - 4 T^{15} + \cdots + 1 \) Copy content Toggle raw display
$17$ \( T^{16} - 2 T^{14} + \cdots + 1 \) Copy content Toggle raw display
$19$ \( T^{16} \) Copy content Toggle raw display
$23$ \( T^{16} \) Copy content Toggle raw display
$29$ \( T^{16} - 4 T^{15} + \cdots + 16 \) Copy content Toggle raw display
$31$ \( T^{16} \) Copy content Toggle raw display
$37$ \( T^{16} + 4 T^{14} + \cdots + 1 \) Copy content Toggle raw display
$41$ \( T^{16} - T^{12} + \cdots + 1 \) Copy content Toggle raw display
$43$ \( T^{16} \) Copy content Toggle raw display
$47$ \( T^{16} \) Copy content Toggle raw display
$53$ \( T^{16} - 2 T^{14} + \cdots + 1 \) Copy content Toggle raw display
$59$ \( T^{16} \) Copy content Toggle raw display
$61$ \( (T^{8} - 2 T^{7} + 2 T^{6} + \cdots + 1)^{2} \) Copy content Toggle raw display
$67$ \( T^{16} \) Copy content Toggle raw display
$71$ \( T^{16} \) Copy content Toggle raw display
$73$ \( (T^{8} + 7 T^{4} + 1)^{2} \) Copy content Toggle raw display
$79$ \( T^{16} \) Copy content Toggle raw display
$83$ \( T^{16} \) Copy content Toggle raw display
$89$ \( T^{16} - 2 T^{14} + \cdots + 16 \) Copy content Toggle raw display
$97$ \( T^{16} - 2 T^{14} + \cdots + 1 \) Copy content Toggle raw display
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