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Properties

Label 3640.1.fq.a
Level $3640$
Weight $1$
Character orbit 3640.fq
Analytic conductor $1.817$
Analytic rank $0$
Dimension $2$
Projective image $D_{3}$
CM discriminant -40
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3640,1,Mod(3019,3640)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3640, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 3, 3, 2, 2]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3640.3019");
 
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.fq (of order \(6\), degree \(2\), 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: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{3}\)
Projective field: Galois closure of 3.1.331240.2
Artin image: $C_6\times S_3$
Artin field: Galois closure of \(\mathbb{Q}[x]/(x^{12} - \cdots)\)

$q$-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.

\(f(q)\) \(=\) \( q - q^{2} + q^{4} + \zeta_{6} q^{5} - q^{7} - q^{8} + \zeta_{6}^{2} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} + q^{4} + \zeta_{6} q^{5} - q^{7} - q^{8} + \zeta_{6}^{2} q^{9} - \zeta_{6} q^{10} - \zeta_{6} q^{11} - q^{13} + q^{14} + q^{16} - \zeta_{6}^{2} q^{18} - \zeta_{6}^{2} q^{19} + \zeta_{6} q^{20} + 2 \zeta_{6} q^{22} + q^{23} + \zeta_{6}^{2} q^{25} + q^{26} - q^{28} - q^{32} - \zeta_{6} q^{35} + \zeta_{6}^{2} q^{36} + q^{37} + \zeta_{6}^{2} q^{38} - \zeta_{6} q^{40} - \zeta_{6}^{2} q^{41} - 2 \zeta_{6} q^{44} - q^{45} - q^{46} - \zeta_{6} q^{47} + q^{49} - \zeta_{6}^{2} q^{50} - q^{52} - \zeta_{6}^{2} q^{53} - 2 \zeta_{6}^{2} q^{55} + q^{56} + q^{59} - \zeta_{6}^{2} q^{63} + q^{64} - \zeta_{6} q^{65} + \zeta_{6} q^{70} - \zeta_{6}^{2} q^{72} - q^{74} - \zeta_{6}^{2} q^{76} + 2 \zeta_{6} q^{77} + \zeta_{6} q^{80} - \zeta_{6} q^{81} + \zeta_{6}^{2} q^{82} + 2 \zeta_{6} q^{88} - q^{89} + q^{90} + q^{91} + q^{92} + \zeta_{6} q^{94} + q^{95} - q^{98} + 2 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + 2 q^{4} + q^{5} - 2 q^{7} - 2 q^{8} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} + 2 q^{4} + q^{5} - 2 q^{7} - 2 q^{8} - q^{9} - q^{10} - 2 q^{11} - 2 q^{13} + 2 q^{14} + 2 q^{16} + q^{18} + q^{19} + q^{20} + 2 q^{22} + 2 q^{23} - q^{25} + 2 q^{26} - 2 q^{28} - 2 q^{32} - q^{35} - q^{36} + 2 q^{37} - q^{38} - q^{40} + q^{41} - 2 q^{44} - 2 q^{45} - 2 q^{46} - q^{47} + 2 q^{49} + q^{50} - 2 q^{52} + 2 q^{53} + 2 q^{55} + 2 q^{56} + 4 q^{59} + q^{63} + 2 q^{64} - q^{65} + q^{70} + q^{72} - 2 q^{74} + q^{76} + 2 q^{77} + q^{80} - q^{81} - q^{82} + 2 q^{88} - 2 q^{89} + 2 q^{90} + 2 q^{91} + 2 q^{92} + q^{94} + 2 q^{95} - 2 q^{98} + 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

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

\(n\) \(521\) \(561\) \(911\) \(1457\) \(1821\)
\(\chi(n)\) \(-\zeta_{6}\) \(-\zeta_{6}\) \(-1\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3019.1
0.500000 − 0.866025i
0.500000 + 0.866025i
−1.00000 0 1.00000 0.500000 − 0.866025i 0 −1.00000 −1.00000 −0.500000 − 0.866025i −0.500000 + 0.866025i
3259.1 −1.00000 0 1.00000 0.500000 + 0.866025i 0 −1.00000 −1.00000 −0.500000 + 0.866025i −0.500000 − 0.866025i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
40.e odd 2 1 CM by \(\Q(\sqrt{-10}) \)
91.h even 3 1 inner
3640.fq odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3640.1.fq.a yes 2
5.b even 2 1 3640.1.fq.d yes 2
7.c even 3 1 3640.1.du.d yes 2
8.d odd 2 1 3640.1.fq.d yes 2
13.c even 3 1 3640.1.du.d yes 2
35.j even 6 1 3640.1.du.a ✓ 2
40.e odd 2 1 CM 3640.1.fq.a yes 2
56.k odd 6 1 3640.1.du.a ✓ 2
65.n even 6 1 3640.1.du.a ✓ 2
91.h even 3 1 inner 3640.1.fq.a yes 2
104.n odd 6 1 3640.1.du.a ✓ 2
280.bi odd 6 1 3640.1.du.d yes 2
455.ba even 6 1 3640.1.fq.d yes 2
520.bx odd 6 1 3640.1.du.d yes 2
728.bx odd 6 1 3640.1.fq.d yes 2
3640.fq odd 6 1 inner 3640.1.fq.a yes 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3640.1.du.a ✓ 2 35.j even 6 1
3640.1.du.a ✓ 2 56.k odd 6 1
3640.1.du.a ✓ 2 65.n even 6 1
3640.1.du.a ✓ 2 104.n odd 6 1
3640.1.du.d yes 2 7.c even 3 1
3640.1.du.d yes 2 13.c even 3 1
3640.1.du.d yes 2 280.bi odd 6 1
3640.1.du.d yes 2 520.bx odd 6 1
3640.1.fq.a yes 2 1.a even 1 1 trivial
3640.1.fq.a yes 2 40.e odd 2 1 CM
3640.1.fq.a yes 2 91.h even 3 1 inner
3640.1.fq.a yes 2 3640.fq odd 6 1 inner
3640.1.fq.d yes 2 5.b even 2 1
3640.1.fq.d yes 2 8.d odd 2 1
3640.1.fq.d yes 2 455.ba even 6 1
3640.1.fq.d yes 2 728.bx odd 6 1

Hecke kernels

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

\( T_{11}^{2} + 2T_{11} + 4 \) Copy content Toggle raw display
\( T_{23} - 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

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