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Properties

Label 3640.1.n.g
Level $3640$
Weight $1$
Character orbit 3640.n
Analytic conductor $1.817$
Analytic rank $0$
Dimension $8$
Projective image $D_{10}$
CM discriminant -455
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3640,1,Mod(909,3640)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3640, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 1, 1, 1]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3640.909");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3640 = 2^{3} \cdot 5 \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3640.n (of order \(2\), degree \(1\), 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: \(1.81659664598\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\Q(\zeta_{20})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{6} + x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{10}\)
Projective field: Galois closure of 10.0.9830906408960000.1

$q$-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.

\(f(q)\) \(=\) \( q - \zeta_{20}^{2} q^{2} + ( - \zeta_{20}^{9} - \zeta_{20}) q^{3} + \zeta_{20}^{4} q^{4} - \zeta_{20}^{5} q^{5} + (\zeta_{20}^{3} - \zeta_{20}) q^{6} - q^{7} - \zeta_{20}^{6} q^{8} + ( - \zeta_{20}^{8} + \zeta_{20}^{2} - 1) q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - \zeta_{20}^{2} q^{2} + ( - \zeta_{20}^{9} - \zeta_{20}) q^{3} + \zeta_{20}^{4} q^{4} - \zeta_{20}^{5} q^{5} + (\zeta_{20}^{3} - \zeta_{20}) q^{6} - q^{7} - \zeta_{20}^{6} q^{8} + ( - \zeta_{20}^{8} + \zeta_{20}^{2} - 1) q^{9} + \zeta_{20}^{7} q^{10} + ( - \zeta_{20}^{5} + \zeta_{20}^{3}) q^{12} + \zeta_{20}^{5} q^{13} + \zeta_{20}^{2} q^{14} + (\zeta_{20}^{6} - \zeta_{20}^{4}) q^{15} + \zeta_{20}^{8} q^{16} + ( - \zeta_{20}^{7} + \zeta_{20}^{3}) q^{17} + ( - \zeta_{20}^{4} + \zeta_{20}^{2} - 1) q^{18} + (\zeta_{20}^{7} + \zeta_{20}^{3}) q^{19} - \zeta_{20}^{9} q^{20} + (\zeta_{20}^{9} + \zeta_{20}) q^{21} + (\zeta_{20}^{7} - \zeta_{20}^{5}) q^{24} - q^{25} - \zeta_{20}^{7} q^{26} + (\zeta_{20}^{9} - \zeta_{20}^{7} + \cdots + \zeta_{20}) q^{27} + \cdots - \zeta_{20}^{2} q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 2 q^{2} - 2 q^{4} - 8 q^{7} - 2 q^{8} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 2 q^{2} - 2 q^{4} - 8 q^{7} - 2 q^{8} - 4 q^{9} + 2 q^{14} + 4 q^{15} - 2 q^{16} - 4 q^{18} - 8 q^{25} + 2 q^{28} + 4 q^{30} + 8 q^{32} + 6 q^{36} - 4 q^{39} + 8 q^{49} + 2 q^{50} + 2 q^{56} + 8 q^{57} - 6 q^{60} + 4 q^{63} - 2 q^{64} + 8 q^{65} + 6 q^{72} - 4 q^{78} - 4 q^{79} + 4 q^{95} - 2 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3640\mathbb{Z}\right)^\times\).

\(n\) \(521\) \(561\) \(911\) \(1457\) \(1821\)
\(\chi(n)\) \(-1\) \(-1\) \(1\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
909.1
0.951057 + 0.309017i
−0.951057 − 0.309017i
−0.951057 + 0.309017i
0.951057 − 0.309017i
0.587785 + 0.809017i
−0.587785 − 0.809017i
−0.587785 + 0.809017i
0.587785 − 0.809017i
−0.809017 − 0.587785i − 0.618034i 0.309017 + 0.951057i − 1.00000i −0.363271 + 0.500000i −1.00000 0.309017 − 0.951057i 0.618034 −0.587785 + 0.809017i
909.2 −0.809017 − 0.587785i 0.618034i 0.309017 + 0.951057i 1.00000i 0.363271 − 0.500000i −1.00000 0.309017 − 0.951057i 0.618034 0.587785 − 0.809017i
909.3 −0.809017 + 0.587785i − 0.618034i 0.309017 − 0.951057i − 1.00000i 0.363271 + 0.500000i −1.00000 0.309017 + 0.951057i 0.618034 0.587785 + 0.809017i
909.4 −0.809017 + 0.587785i 0.618034i 0.309017 − 0.951057i 1.00000i −0.363271 − 0.500000i −1.00000 0.309017 + 0.951057i 0.618034 −0.587785 − 0.809017i
909.5 0.309017 − 0.951057i − 1.61803i −0.809017 − 0.587785i 1.00000i −1.53884 − 0.500000i −1.00000 −0.809017 + 0.587785i −1.61803 0.951057 + 0.309017i
909.6 0.309017 − 0.951057i 1.61803i −0.809017 − 0.587785i − 1.00000i 1.53884 + 0.500000i −1.00000 −0.809017 + 0.587785i −1.61803 −0.951057 − 0.309017i
909.7 0.309017 + 0.951057i − 1.61803i −0.809017 + 0.587785i 1.00000i 1.53884 − 0.500000i −1.00000 −0.809017 − 0.587785i −1.61803 −0.951057 + 0.309017i
909.8 0.309017 + 0.951057i 1.61803i −0.809017 + 0.587785i − 1.00000i −1.53884 + 0.500000i −1.00000 −0.809017 − 0.587785i −1.61803 0.951057 − 0.309017i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 909.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
455.h odd 2 1 CM by \(\Q(\sqrt{-455}) \)
7.b odd 2 1 inner
8.b even 2 1 inner
56.h odd 2 1 inner
65.d even 2 1 inner
520.p even 2 1 inner
3640.n odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3640.1.n.g ✓ 8
5.b even 2 1 3640.1.n.h yes 8
7.b odd 2 1 inner 3640.1.n.g ✓ 8
8.b even 2 1 inner 3640.1.n.g ✓ 8
13.b even 2 1 3640.1.n.h yes 8
35.c odd 2 1 3640.1.n.h yes 8
40.f even 2 1 3640.1.n.h yes 8
56.h odd 2 1 inner 3640.1.n.g ✓ 8
65.d even 2 1 inner 3640.1.n.g ✓ 8
91.b odd 2 1 3640.1.n.h yes 8
104.e even 2 1 3640.1.n.h yes 8
280.c odd 2 1 3640.1.n.h yes 8
455.h odd 2 1 CM 3640.1.n.g ✓ 8
520.p even 2 1 inner 3640.1.n.g ✓ 8
728.l odd 2 1 3640.1.n.h yes 8
3640.n odd 2 1 inner 3640.1.n.g ✓ 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3640.1.n.g ✓ 8 1.a even 1 1 trivial
3640.1.n.g ✓ 8 7.b odd 2 1 inner
3640.1.n.g ✓ 8 8.b even 2 1 inner
3640.1.n.g ✓ 8 56.h odd 2 1 inner
3640.1.n.g ✓ 8 65.d even 2 1 inner
3640.1.n.g ✓ 8 455.h odd 2 1 CM
3640.1.n.g ✓ 8 520.p even 2 1 inner
3640.1.n.g ✓ 8 3640.n odd 2 1 inner
3640.1.n.h yes 8 5.b even 2 1
3640.1.n.h yes 8 13.b even 2 1
3640.1.n.h yes 8 35.c odd 2 1
3640.1.n.h yes 8 40.f even 2 1
3640.1.n.h yes 8 91.b odd 2 1
3640.1.n.h yes 8 104.e even 2 1
3640.1.n.h yes 8 280.c odd 2 1
3640.1.n.h yes 8 728.l odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(3640, [\chi])\):

\( T_{3}^{4} + 3T_{3}^{2} + 1 \) Copy content Toggle raw display
\( T_{61} \) Copy content Toggle raw display
\( T_{83} \) Copy content Toggle raw display
\( T_{137}^{2} + T_{137} - 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

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