Properties

Label 3648.1.bl.a
Level $3648$
Weight $1$
Character orbit 3648.bl
Analytic conductor $1.821$
Analytic rank $0$
Dimension $2$
Projective image $D_{3}$
CM discriminant -3
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3648,1,Mod(1793,3648)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3648, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 3, 2]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3648.1793");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3648 = 2^{6} \cdot 3 \cdot 19 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3648.bl (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: \(1.82058916609\)
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: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 57)
Projective image: \(D_{3}\)
Projective field: Galois closure of 3.1.1083.1
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.

\(f(q)\) \(=\) \( q + \zeta_{6}^{2} q^{3} + q^{7} - \zeta_{6} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{6}^{2} q^{3} + q^{7} - \zeta_{6} q^{9} - \zeta_{6} q^{13} + q^{19} + \zeta_{6}^{2} q^{21} - \zeta_{6} q^{25} + q^{27} + q^{31} + q^{37} + q^{39} - \zeta_{6}^{2} q^{43} + \zeta_{6}^{2} q^{57} - \zeta_{6} q^{61} - \zeta_{6} q^{63} + \zeta_{6} q^{67} - \zeta_{6}^{2} q^{73} + q^{75} + \zeta_{6}^{2} q^{79} + \zeta_{6}^{2} q^{81} - \zeta_{6} q^{91} + \zeta_{6}^{2} q^{93} + \zeta_{6}^{2} q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{3} + 2 q^{7} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - q^{3} + 2 q^{7} - q^{9} - q^{13} + 2 q^{19} - q^{21} - q^{25} + 2 q^{27} + 2 q^{31} + 2 q^{37} + 2 q^{39} + q^{43} - q^{57} - q^{61} - q^{63} + q^{67} + q^{73} + 2 q^{75} - q^{79} - q^{81} - q^{91} - q^{93} - 2 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3648\mathbb{Z}\right)^\times\).

\(n\) \(1217\) \(1921\) \(2053\) \(2623\)
\(\chi(n)\) \(-1\) \(-\zeta_{6}\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1793.1
0.500000 0.866025i
0.500000 + 0.866025i
0 −0.500000 0.866025i 0 0 0 1.00000 0 −0.500000 + 0.866025i 0
2177.1 0 −0.500000 + 0.866025i 0 0 0 1.00000 0 −0.500000 0.866025i 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)
19.c even 3 1 inner
57.h odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3648.1.bl.a 2
3.b odd 2 1 CM 3648.1.bl.a 2
4.b odd 2 1 3648.1.bl.b 2
8.b even 2 1 912.1.bl.a 2
8.d odd 2 1 57.1.h.a 2
12.b even 2 1 3648.1.bl.b 2
19.c even 3 1 inner 3648.1.bl.a 2
24.f even 2 1 57.1.h.a 2
24.h odd 2 1 912.1.bl.a 2
40.e odd 2 1 1425.1.t.a 2
40.k even 4 2 1425.1.o.a 4
56.e even 2 1 2793.1.bf.a 2
56.k odd 6 1 2793.1.n.a 2
56.k odd 6 1 2793.1.bi.b 2
56.m even 6 1 2793.1.n.b 2
56.m even 6 1 2793.1.bi.a 2
57.h odd 6 1 inner 3648.1.bl.a 2
72.l even 6 1 1539.1.j.a 2
72.l even 6 1 1539.1.n.a 2
72.p odd 6 1 1539.1.j.a 2
72.p odd 6 1 1539.1.n.a 2
76.g odd 6 1 3648.1.bl.b 2
120.m even 2 1 1425.1.t.a 2
120.q odd 4 2 1425.1.o.a 4
152.b even 2 1 1083.1.h.a 2
152.k odd 6 1 57.1.h.a 2
152.k odd 6 1 1083.1.b.b 1
152.o even 6 1 1083.1.b.a 1
152.o even 6 1 1083.1.h.a 2
152.p even 6 1 912.1.bl.a 2
152.u odd 18 6 1083.1.l.a 6
152.v even 18 6 1083.1.l.b 6
168.e odd 2 1 2793.1.bf.a 2
168.v even 6 1 2793.1.n.a 2
168.v even 6 1 2793.1.bi.b 2
168.be odd 6 1 2793.1.n.b 2
168.be odd 6 1 2793.1.bi.a 2
228.m even 6 1 3648.1.bl.b 2
456.l odd 2 1 1083.1.h.a 2
456.s odd 6 1 1083.1.b.a 1
456.s odd 6 1 1083.1.h.a 2
456.u even 6 1 57.1.h.a 2
456.u even 6 1 1083.1.b.b 1
456.x odd 6 1 912.1.bl.a 2
456.bt odd 18 6 1083.1.l.b 6
456.bu even 18 6 1083.1.l.a 6
760.bm odd 6 1 1425.1.t.a 2
760.bw even 12 2 1425.1.o.a 4
1064.u even 6 1 2793.1.n.b 2
1064.bd odd 6 1 2793.1.bi.b 2
1064.ch even 6 1 2793.1.bf.a 2
1064.cr even 6 1 2793.1.bi.a 2
1064.cz odd 6 1 2793.1.n.a 2
1368.ba odd 6 1 1539.1.j.a 2
1368.bn even 6 1 1539.1.n.a 2
1368.cj odd 6 1 1539.1.n.a 2
1368.db even 6 1 1539.1.j.a 2
2280.co even 6 1 1425.1.t.a 2
2280.dj odd 12 2 1425.1.o.a 4
3192.bs odd 6 1 2793.1.bi.a 2
3192.df even 6 1 2793.1.n.a 2
3192.du odd 6 1 2793.1.bf.a 2
3192.ex odd 6 1 2793.1.n.b 2
3192.gd even 6 1 2793.1.bi.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
57.1.h.a 2 8.d odd 2 1
57.1.h.a 2 24.f even 2 1
57.1.h.a 2 152.k odd 6 1
57.1.h.a 2 456.u even 6 1
912.1.bl.a 2 8.b even 2 1
912.1.bl.a 2 24.h odd 2 1
912.1.bl.a 2 152.p even 6 1
912.1.bl.a 2 456.x odd 6 1
1083.1.b.a 1 152.o even 6 1
1083.1.b.a 1 456.s odd 6 1
1083.1.b.b 1 152.k odd 6 1
1083.1.b.b 1 456.u even 6 1
1083.1.h.a 2 152.b even 2 1
1083.1.h.a 2 152.o even 6 1
1083.1.h.a 2 456.l odd 2 1
1083.1.h.a 2 456.s odd 6 1
1083.1.l.a 6 152.u odd 18 6
1083.1.l.a 6 456.bu even 18 6
1083.1.l.b 6 152.v even 18 6
1083.1.l.b 6 456.bt odd 18 6
1425.1.o.a 4 40.k even 4 2
1425.1.o.a 4 120.q odd 4 2
1425.1.o.a 4 760.bw even 12 2
1425.1.o.a 4 2280.dj odd 12 2
1425.1.t.a 2 40.e odd 2 1
1425.1.t.a 2 120.m even 2 1
1425.1.t.a 2 760.bm odd 6 1
1425.1.t.a 2 2280.co even 6 1
1539.1.j.a 2 72.l even 6 1
1539.1.j.a 2 72.p odd 6 1
1539.1.j.a 2 1368.ba odd 6 1
1539.1.j.a 2 1368.db even 6 1
1539.1.n.a 2 72.l even 6 1
1539.1.n.a 2 72.p odd 6 1
1539.1.n.a 2 1368.bn even 6 1
1539.1.n.a 2 1368.cj odd 6 1
2793.1.n.a 2 56.k odd 6 1
2793.1.n.a 2 168.v even 6 1
2793.1.n.a 2 1064.cz odd 6 1
2793.1.n.a 2 3192.df even 6 1
2793.1.n.b 2 56.m even 6 1
2793.1.n.b 2 168.be odd 6 1
2793.1.n.b 2 1064.u even 6 1
2793.1.n.b 2 3192.ex odd 6 1
2793.1.bf.a 2 56.e even 2 1
2793.1.bf.a 2 168.e odd 2 1
2793.1.bf.a 2 1064.ch even 6 1
2793.1.bf.a 2 3192.du odd 6 1
2793.1.bi.a 2 56.m even 6 1
2793.1.bi.a 2 168.be odd 6 1
2793.1.bi.a 2 1064.cr even 6 1
2793.1.bi.a 2 3192.bs odd 6 1
2793.1.bi.b 2 56.k odd 6 1
2793.1.bi.b 2 168.v even 6 1
2793.1.bi.b 2 1064.bd odd 6 1
2793.1.bi.b 2 3192.gd even 6 1
3648.1.bl.a 2 1.a even 1 1 trivial
3648.1.bl.a 2 3.b odd 2 1 CM
3648.1.bl.a 2 19.c even 3 1 inner
3648.1.bl.a 2 57.h odd 6 1 inner
3648.1.bl.b 2 4.b odd 2 1
3648.1.bl.b 2 12.b even 2 1
3648.1.bl.b 2 76.g odd 6 1
3648.1.bl.b 2 228.m even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7} - 1 \) acting on \(S_{1}^{\mathrm{new}}(3648, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

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