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Properties

Label 3660.1.eb.e
Level $3660$
Weight $1$
Character orbit 3660.eb
Analytic conductor $1.827$
Analytic rank $0$
Dimension $16$
Projective image $D_{20}$
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] = [3660,1,Mod(419,3660)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3660, base_ring=CyclotomicField(20))
 
chi = DirichletCharacter(H, H._module([10, 10, 10, 11]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3660.419");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3660 = 2^{2} \cdot 3 \cdot 5 \cdot 61 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3660.eb (of order \(20\), degree \(8\), 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: \(1.82657794624\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(2\) over \(\Q(\zeta_{20})\)
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: \(D_{20}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{20} - \cdots)\)

$q$-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.

\(f(q)\) \(=\) \( q - \zeta_{40}^{11} q^{2} - \zeta_{40}^{15} q^{3} - \zeta_{40}^{2} q^{4} - \zeta_{40}^{12} q^{5} - \zeta_{40}^{6} q^{6} + ( - \zeta_{40}^{17} - \zeta_{40}) q^{7} + \zeta_{40}^{13} q^{8} - \zeta_{40}^{10} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - \zeta_{40}^{11} q^{2} - \zeta_{40}^{15} q^{3} - \zeta_{40}^{2} q^{4} - \zeta_{40}^{12} q^{5} - \zeta_{40}^{6} q^{6} + ( - \zeta_{40}^{17} - \zeta_{40}) q^{7} + \zeta_{40}^{13} q^{8} - \zeta_{40}^{10} q^{9} - \zeta_{40}^{3} q^{10} + \zeta_{40}^{17} q^{12} + (\zeta_{40}^{12} - \zeta_{40}^{8}) q^{14} - \zeta_{40}^{7} q^{15} + \zeta_{40}^{4} q^{16} - \zeta_{40} q^{18} + \zeta_{40}^{14} q^{20} + (\zeta_{40}^{16} - \zeta_{40}^{12}) q^{21} + (\zeta_{40}^{13} + \zeta_{40}) q^{23} + \zeta_{40}^{8} q^{24} - \zeta_{40}^{4} q^{25} - \zeta_{40}^{5} q^{27} + (\zeta_{40}^{19} + \zeta_{40}^{3}) q^{28} + ( - \zeta_{40}^{8} - \zeta_{40}^{2}) q^{29} + \zeta_{40}^{18} q^{30} - \zeta_{40}^{15} q^{32} + (\zeta_{40}^{13} - \zeta_{40}^{9}) q^{35} + \zeta_{40}^{12} q^{36} + \zeta_{40}^{5} q^{40} + ( - \zeta_{40}^{8} - 1) q^{41} + (\zeta_{40}^{7} - \zeta_{40}^{3}) q^{42} + ( - \zeta_{40}^{15} + \zeta_{40}^{11}) q^{43} - \zeta_{40}^{2} q^{45} + ( - \zeta_{40}^{12} + \zeta_{40}^{4}) q^{46} + ( - \zeta_{40}^{19} - \zeta_{40}) q^{47} - \zeta_{40}^{19} q^{48} + (\zeta_{40}^{18} + \cdots + \zeta_{40}^{2}) q^{49} + \cdots + ( - \zeta_{40}^{13} + \cdots - \zeta_{40}^{5}) q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 4 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 4 q^{5} + 8 q^{14} + 4 q^{16} - 8 q^{21} - 4 q^{24} - 4 q^{25} + 4 q^{29} + 4 q^{36} - 12 q^{41} - 4 q^{54} - 12 q^{70} + 4 q^{80} - 16 q^{81} + 4 q^{89} + 4 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3660\mathbb{Z}\right)^\times\).

\(n\) \(1831\) \(2197\) \(2441\) \(3601\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\) \(\zeta_{40}^{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
419.1
−0.156434 + 0.987688i
0.156434 − 0.987688i
−0.156434 − 0.987688i
0.156434 + 0.987688i
0.453990 − 0.891007i
−0.453990 + 0.891007i
0.453990 + 0.891007i
−0.453990 − 0.891007i
0.891007 − 0.453990i
−0.891007 + 0.453990i
0.891007 + 0.453990i
−0.891007 − 0.453990i
−0.987688 + 0.156434i
0.987688 − 0.156434i
−0.987688 − 0.156434i
0.987688 + 0.156434i
−0.987688 − 0.156434i −0.707107 − 0.707107i 0.951057 + 0.309017i 0.309017 − 0.951057i 0.587785 + 0.809017i 0.610425 − 0.0966818i −0.891007 − 0.453990i 1.00000i −0.453990 + 0.891007i
419.2 0.987688 + 0.156434i 0.707107 + 0.707107i 0.951057 + 0.309017i 0.309017 − 0.951057i 0.587785 + 0.809017i −0.610425 + 0.0966818i 0.891007 + 0.453990i 1.00000i 0.453990 − 0.891007i
1319.1 −0.987688 + 0.156434i −0.707107 + 0.707107i 0.951057 − 0.309017i 0.309017 + 0.951057i 0.587785 − 0.809017i 0.610425 + 0.0966818i −0.891007 + 0.453990i − 1.00000i −0.453990 − 0.891007i
1319.2 0.987688 − 0.156434i 0.707107 − 0.707107i 0.951057 − 0.309017i 0.309017 + 0.951057i 0.587785 − 0.809017i −0.610425 − 0.0966818i 0.891007 − 0.453990i − 1.00000i 0.453990 + 0.891007i
1379.1 −0.891007 − 0.453990i 0.707107 − 0.707107i 0.587785 + 0.809017i −0.809017 + 0.587785i −0.951057 + 0.309017i −1.44168 + 0.734572i −0.156434 − 0.987688i − 1.00000i 0.987688 − 0.156434i
1379.2 0.891007 + 0.453990i −0.707107 + 0.707107i 0.587785 + 0.809017i −0.809017 + 0.587785i −0.951057 + 0.309017i 1.44168 − 0.734572i 0.156434 + 0.987688i − 1.00000i −0.987688 + 0.156434i
1619.1 −0.891007 + 0.453990i 0.707107 + 0.707107i 0.587785 − 0.809017i −0.809017 − 0.587785i −0.951057 − 0.309017i −1.44168 − 0.734572i −0.156434 + 0.987688i 1.00000i 0.987688 + 0.156434i
1619.2 0.891007 − 0.453990i −0.707107 − 0.707107i 0.587785 − 0.809017i −0.809017 − 0.587785i −0.951057 − 0.309017i 1.44168 + 0.734572i 0.156434 − 0.987688i 1.00000i −0.987688 − 0.156434i
1919.1 −0.453990 − 0.891007i −0.707107 + 0.707107i −0.587785 + 0.809017i −0.809017 − 0.587785i 0.951057 + 0.309017i −0.734572 + 1.44168i 0.987688 + 0.156434i − 1.00000i −0.156434 + 0.987688i
1919.2 0.453990 + 0.891007i 0.707107 − 0.707107i −0.587785 + 0.809017i −0.809017 − 0.587785i 0.951057 + 0.309017i 0.734572 − 1.44168i −0.987688 − 0.156434i − 1.00000i 0.156434 − 0.987688i
2159.1 −0.453990 + 0.891007i −0.707107 − 0.707107i −0.587785 − 0.809017i −0.809017 + 0.587785i 0.951057 − 0.309017i −0.734572 − 1.44168i 0.987688 − 0.156434i 1.00000i −0.156434 − 0.987688i
2159.2 0.453990 − 0.891007i 0.707107 + 0.707107i −0.587785 − 0.809017i −0.809017 + 0.587785i 0.951057 − 0.309017i 0.734572 + 1.44168i −0.987688 + 0.156434i 1.00000i 0.156434 + 0.987688i
2219.1 −0.156434 − 0.987688i −0.707107 − 0.707107i −0.951057 + 0.309017i 0.309017 + 0.951057i −0.587785 + 0.809017i 0.0966818 − 0.610425i 0.453990 + 0.891007i 1.00000i 0.891007 − 0.453990i
2219.2 0.156434 + 0.987688i 0.707107 + 0.707107i −0.951057 + 0.309017i 0.309017 + 0.951057i −0.587785 + 0.809017i −0.0966818 + 0.610425i −0.453990 − 0.891007i 1.00000i −0.891007 + 0.453990i
3119.1 −0.156434 + 0.987688i −0.707107 + 0.707107i −0.951057 − 0.309017i 0.309017 − 0.951057i −0.587785 − 0.809017i 0.0966818 + 0.610425i 0.453990 − 0.891007i − 1.00000i 0.891007 + 0.453990i
3119.2 0.156434 − 0.987688i 0.707107 − 0.707107i −0.951057 − 0.309017i 0.309017 − 0.951057i −0.587785 − 0.809017i −0.0966818 − 0.610425i −0.453990 + 0.891007i − 1.00000i −0.891007 − 0.453990i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 419.2
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
183.r even 20 1 inner
732.bh odd 20 1 inner
915.cc even 20 1 inner
3660.eb odd 20 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3660.1.eb.e ✓ 16
3.b odd 2 1 3660.1.eb.f yes 16
4.b odd 2 1 inner 3660.1.eb.e ✓ 16
5.b even 2 1 inner 3660.1.eb.e ✓ 16
12.b even 2 1 3660.1.eb.f yes 16
15.d odd 2 1 3660.1.eb.f yes 16
20.d odd 2 1 CM 3660.1.eb.e ✓ 16
60.h even 2 1 3660.1.eb.f yes 16
61.j odd 20 1 3660.1.eb.f yes 16
183.r even 20 1 inner 3660.1.eb.e ✓ 16
244.r even 20 1 3660.1.eb.f yes 16
305.ba odd 20 1 3660.1.eb.f yes 16
732.bh odd 20 1 inner 3660.1.eb.e ✓ 16
915.cc even 20 1 inner 3660.1.eb.e ✓ 16
1220.ch even 20 1 3660.1.eb.f yes 16
3660.eb odd 20 1 inner 3660.1.eb.e ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3660.1.eb.e ✓ 16 1.a even 1 1 trivial
3660.1.eb.e ✓ 16 4.b odd 2 1 inner
3660.1.eb.e ✓ 16 5.b even 2 1 inner
3660.1.eb.e ✓ 16 20.d odd 2 1 CM
3660.1.eb.e ✓ 16 183.r even 20 1 inner
3660.1.eb.e ✓ 16 732.bh odd 20 1 inner
3660.1.eb.e ✓ 16 915.cc even 20 1 inner
3660.1.eb.e ✓ 16 3660.eb odd 20 1 inner
3660.1.eb.f yes 16 3.b odd 2 1
3660.1.eb.f yes 16 12.b even 2 1
3660.1.eb.f yes 16 15.d odd 2 1
3660.1.eb.f yes 16 60.h even 2 1
3660.1.eb.f yes 16 61.j odd 20 1
3660.1.eb.f yes 16 244.r even 20 1
3660.1.eb.f yes 16 305.ba odd 20 1
3660.1.eb.f yes 16 1220.ch even 20 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(3660, [\chi])\):

\( T_{7}^{16} + 4T_{7}^{12} + 46T_{7}^{8} - 11T_{7}^{4} + 1 \) Copy content Toggle raw display
\( T_{17} \) Copy content Toggle raw display
\( T_{23}^{16} - 20T_{23}^{12} + 150T_{23}^{8} + 125T_{23}^{4} + 625 \) Copy content Toggle raw display
\( T_{29}^{8} - 2T_{29}^{7} + 2T_{29}^{6} + 11T_{29}^{4} - 20T_{29}^{3} + 18T_{29}^{2} - 6T_{29} + 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} - T^{12} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( (T^{4} + 1)^{4} \) Copy content Toggle raw display
$5$ \( (T^{4} + T^{3} + T^{2} + \cdots + 1)^{4} \) Copy content Toggle raw display
$7$ \( T^{16} + 4 T^{12} + \cdots + 1 \) Copy content Toggle raw display
$11$ \( T^{16} \) Copy content Toggle raw display
$13$ \( T^{16} \) Copy content Toggle raw display
$17$ \( T^{16} \) Copy content Toggle raw display
$19$ \( T^{16} \) Copy content Toggle raw display
$23$ \( T^{16} - 20 T^{12} + \cdots + 625 \) Copy content Toggle raw display
$29$ \( (T^{8} - 2 T^{7} + 2 T^{6} + \cdots + 1)^{2} \) Copy content Toggle raw display
$31$ \( T^{16} \) Copy content Toggle raw display
$37$ \( T^{16} \) Copy content Toggle raw display
$41$ \( (T^{4} + 3 T^{3} + 4 T^{2} + \cdots + 1)^{4} \) Copy content Toggle raw display
$43$ \( T^{16} - 11 T^{12} + \cdots + 1 \) Copy content Toggle raw display
$47$ \( (T^{8} + 8 T^{6} + 19 T^{4} + \cdots + 1)^{2} \) Copy content Toggle raw display
$53$ \( T^{16} \) Copy content Toggle raw display
$59$ \( T^{16} \) Copy content Toggle raw display
$61$ \( (T^{8} - T^{6} + T^{4} + \cdots + 1)^{2} \) Copy content Toggle raw display
$67$ \( T^{16} - 16 T^{12} + \cdots + 65536 \) Copy content Toggle raw display
$71$ \( T^{16} \) Copy content Toggle raw display
$73$ \( T^{16} \) Copy content Toggle raw display
$79$ \( T^{16} \) Copy content Toggle raw display
$83$ \( T^{16} - 4 T^{14} + \cdots + 1 \) Copy content Toggle raw display
$89$ \( (T^{8} - 2 T^{7} + 2 T^{6} + \cdots + 16)^{2} \) Copy content Toggle raw display
$97$ \( T^{16} \) Copy content Toggle raw display
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