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Properties

Label 3240.1.z.i
Level $3240$
Weight $1$
Character orbit 3240.z
Analytic conductor $1.617$
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] = [3240,1,Mod(379,3240)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3240, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 3, 4, 3]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3240.379");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3240 = 2^{3} \cdot 3^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3240.z (of order \(6\), degree \(2\), not 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: \(1.61697064093\)
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: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{3}\)
Projective field: Galois closure of 3.1.3240.1
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.

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

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3240\mathbb{Z}\right)^\times\).

\(n\) \(1297\) \(1621\) \(2431\) \(3161\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\) \(\zeta_{6}^{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
379.1
0.500000 − 0.866025i
0.500000 + 0.866025i
0.500000 − 0.866025i 0 −0.500000 − 0.866025i 0.500000 + 0.866025i 0 0.500000 − 0.866025i −1.00000 0 1.00000
2539.1 0.500000 + 0.866025i 0 −0.500000 + 0.866025i 0.500000 − 0.866025i 0 0.500000 + 0.866025i −1.00000 0 1.00000
\(n\): 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 \(\Q(\sqrt{-10}) \)
9.c even 3 1 inner
360.z odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3240.1.z.i 2
3.b odd 2 1 3240.1.z.c 2
5.b even 2 1 3240.1.z.b 2
8.d odd 2 1 3240.1.z.b 2
9.c even 3 1 3240.1.p.a ✓ 1
9.c even 3 1 inner 3240.1.z.i 2
9.d odd 6 1 3240.1.p.c yes 1
9.d odd 6 1 3240.1.z.c 2
15.d odd 2 1 3240.1.z.h 2
24.f even 2 1 3240.1.z.h 2
40.e odd 2 1 CM 3240.1.z.i 2
45.h odd 6 1 3240.1.p.b yes 1
45.h odd 6 1 3240.1.z.h 2
45.j even 6 1 3240.1.p.d yes 1
45.j even 6 1 3240.1.z.b 2
72.l even 6 1 3240.1.p.b yes 1
72.l even 6 1 3240.1.z.h 2
72.p odd 6 1 3240.1.p.d yes 1
72.p odd 6 1 3240.1.z.b 2
120.m even 2 1 3240.1.z.c 2
360.z odd 6 1 3240.1.p.a ✓ 1
360.z odd 6 1 inner 3240.1.z.i 2
360.bd even 6 1 3240.1.p.c yes 1
360.bd even 6 1 3240.1.z.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3240.1.p.a ✓ 1 9.c even 3 1
3240.1.p.a ✓ 1 360.z odd 6 1
3240.1.p.b yes 1 45.h odd 6 1
3240.1.p.b yes 1 72.l even 6 1
3240.1.p.c yes 1 9.d odd 6 1
3240.1.p.c yes 1 360.bd even 6 1
3240.1.p.d yes 1 45.j even 6 1
3240.1.p.d yes 1 72.p odd 6 1
3240.1.z.b 2 5.b even 2 1
3240.1.z.b 2 8.d odd 2 1
3240.1.z.b 2 45.j even 6 1
3240.1.z.b 2 72.p odd 6 1
3240.1.z.c 2 3.b odd 2 1
3240.1.z.c 2 9.d odd 6 1
3240.1.z.c 2 120.m even 2 1
3240.1.z.c 2 360.bd even 6 1
3240.1.z.h 2 15.d odd 2 1
3240.1.z.h 2 24.f even 2 1
3240.1.z.h 2 45.h odd 6 1
3240.1.z.h 2 72.l even 6 1
3240.1.z.i 2 1.a even 1 1 trivial
3240.1.z.i 2 9.c even 3 1 inner
3240.1.z.i 2 40.e odd 2 1 CM
3240.1.z.i 2 360.z odd 6 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(3240, [\chi])\):

\( T_{7}^{2} - T_{7} + 1 \) Copy content Toggle raw display
\( T_{11}^{2} - 2T_{11} + 4 \) Copy content Toggle raw display
\( T_{17} \) Copy content Toggle raw display
\( T_{23}^{2} + T_{23} + 1 \) Copy content Toggle raw display
\( T_{31} \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$7$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$11$ \( T^{2} - 2T + 4 \) Copy content Toggle raw display
$13$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$17$ \( T^{2} \) Copy content Toggle raw display
$19$ \( (T + 1)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$29$ \( T^{2} \) Copy content Toggle raw display
$31$ \( T^{2} \) Copy content Toggle raw display
$37$ \( (T - 2)^{2} \) 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
$53$ \( (T - 1)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} + T + 1 \) 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
$89$ \( (T + 2)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} \) Copy content Toggle raw display
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