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Properties

Label 3660.1.dh.b
Level $3660$
Weight $1$
Character orbit 3660.dh
Analytic conductor $1.827$
Analytic rank $0$
Dimension $4$
Projective image $D_{12}$
CM discriminant -15
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3660,1,Mod(1199,3660)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3660, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([6, 6, 6, 5]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3660.1199");
 
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.dh (of order \(12\), degree \(4\), 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: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{12}\)
Projective 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.

\(f(q)\) \(=\) \( q - \zeta_{12} q^{2} - q^{3} + \zeta_{12}^{2} q^{4} + \zeta_{12} q^{5} + \zeta_{12} q^{6} - \zeta_{12}^{3} q^{8} + q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - \zeta_{12} q^{2} - q^{3} + \zeta_{12}^{2} q^{4} + \zeta_{12} q^{5} + \zeta_{12} q^{6} - \zeta_{12}^{3} q^{8} + q^{9} - \zeta_{12}^{2} q^{10} - \zeta_{12}^{2} q^{12} - \zeta_{12} q^{15} + \zeta_{12}^{4} q^{16} + ( - \zeta_{12}^{4} + \zeta_{12}^{3}) q^{17} - \zeta_{12} q^{18} + (\zeta_{12}^{3} + \zeta_{12}) q^{19} + \zeta_{12}^{3} q^{20} + (\zeta_{12}^{5} + \zeta_{12}^{4}) q^{23} + \zeta_{12}^{3} q^{24} + \zeta_{12}^{2} q^{25} - q^{27} + \zeta_{12}^{2} q^{30} + ( - \zeta_{12} + 1) q^{31} - \zeta_{12}^{5} q^{32} + (\zeta_{12}^{5} - \zeta_{12}^{4}) q^{34} + \zeta_{12}^{2} q^{36} + ( - \zeta_{12}^{4} - \zeta_{12}^{2}) q^{38} - \zeta_{12}^{4} q^{40} + \zeta_{12} q^{45} + ( - \zeta_{12}^{5} + 1) q^{46} - \zeta_{12}^{5} q^{47} - \zeta_{12}^{4} q^{48} + \zeta_{12}^{5} q^{49} - \zeta_{12}^{3} q^{50} + (\zeta_{12}^{4} - \zeta_{12}^{3}) q^{51} + (\zeta_{12}^{5} - \zeta_{12}^{4}) q^{53} + \zeta_{12} q^{54} + ( - \zeta_{12}^{3} - \zeta_{12}) q^{57} - \zeta_{12}^{3} q^{60} + \zeta_{12}^{5} q^{61} + (\zeta_{12}^{2} - \zeta_{12}) q^{62} - q^{64} + (\zeta_{12}^{5} + 1) q^{68} + ( - \zeta_{12}^{5} - \zeta_{12}^{4}) q^{69} - \zeta_{12}^{3} q^{72} - \zeta_{12}^{2} q^{75} + (\zeta_{12}^{5} + \zeta_{12}^{3}) q^{76} + ( - \zeta_{12}^{3} - \zeta_{12}^{2}) q^{79} + \zeta_{12}^{5} q^{80} + q^{81} + (\zeta_{12}^{2} + 1) q^{83} + ( - \zeta_{12}^{5} + \zeta_{12}^{4}) q^{85} - \zeta_{12}^{2} q^{90} + ( - \zeta_{12} - 1) q^{92} + (\zeta_{12} - 1) q^{93} - q^{94} + (\zeta_{12}^{4} + \zeta_{12}^{2}) q^{95} + \zeta_{12}^{5} q^{96} + q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{3} + 2 q^{4} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{3} + 2 q^{4} + 4 q^{9} - 2 q^{10} - 2 q^{12} - 2 q^{16} + 2 q^{17} - 2 q^{23} + 2 q^{25} - 4 q^{27} + 2 q^{30} + 4 q^{31} + 2 q^{34} + 2 q^{36} + 2 q^{40} + 4 q^{46} + 2 q^{48} - 2 q^{51} + 2 q^{53} + 2 q^{62} - 4 q^{64} + 4 q^{68} + 2 q^{69} - 2 q^{75} - 2 q^{79} + 4 q^{81} + 6 q^{83} - 2 q^{85} - 2 q^{90} - 4 q^{92} - 4 q^{93} - 4 q^{94} + 4 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

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

\(n\) \(1831\) \(2197\) \(2441\) \(3601\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\) \(-\zeta_{12}^{5}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1199.1
−0.866025 − 0.500000i
0.866025 − 0.500000i
−0.866025 + 0.500000i
0.866025 + 0.500000i
0.866025 + 0.500000i −1.00000 0.500000 + 0.866025i −0.866025 − 0.500000i −0.866025 − 0.500000i 0 1.00000i 1.00000 −0.500000 − 0.866025i
1679.1 −0.866025 + 0.500000i −1.00000 0.500000 − 0.866025i 0.866025 − 0.500000i 0.866025 − 0.500000i 0 1.00000i 1.00000 −0.500000 + 0.866025i
1859.1 0.866025 − 0.500000i −1.00000 0.500000 − 0.866025i −0.866025 + 0.500000i −0.866025 + 0.500000i 0 − 1.00000i 1.00000 −0.500000 + 0.866025i
2339.1 −0.866025 − 0.500000i −1.00000 0.500000 + 0.866025i 0.866025 + 0.500000i 0.866025 + 0.500000i 0 − 1.00000i 1.00000 −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
15.d odd 2 1 CM by \(\Q(\sqrt{-15}) \)
244.p even 12 1 inner
3660.dh odd 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3660.1.dh.b yes 4
3.b odd 2 1 3660.1.dh.c yes 4
4.b odd 2 1 3660.1.dh.a ✓ 4
5.b even 2 1 3660.1.dh.c yes 4
12.b even 2 1 3660.1.dh.d yes 4
15.d odd 2 1 CM 3660.1.dh.b yes 4
20.d odd 2 1 3660.1.dh.d yes 4
60.h even 2 1 3660.1.dh.a ✓ 4
61.h odd 12 1 3660.1.dh.a ✓ 4
183.o even 12 1 3660.1.dh.d yes 4
244.p even 12 1 inner 3660.1.dh.b yes 4
305.w odd 12 1 3660.1.dh.d yes 4
732.bc odd 12 1 3660.1.dh.c yes 4
915.bq even 12 1 3660.1.dh.a ✓ 4
1220.bl even 12 1 3660.1.dh.c yes 4
3660.dh odd 12 1 inner 3660.1.dh.b yes 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3660.1.dh.a ✓ 4 4.b odd 2 1
3660.1.dh.a ✓ 4 60.h even 2 1
3660.1.dh.a ✓ 4 61.h odd 12 1
3660.1.dh.a ✓ 4 915.bq even 12 1
3660.1.dh.b yes 4 1.a even 1 1 trivial
3660.1.dh.b yes 4 15.d odd 2 1 CM
3660.1.dh.b yes 4 244.p even 12 1 inner
3660.1.dh.b yes 4 3660.dh odd 12 1 inner
3660.1.dh.c yes 4 3.b odd 2 1
3660.1.dh.c yes 4 5.b even 2 1
3660.1.dh.c yes 4 732.bc odd 12 1
3660.1.dh.c yes 4 1220.bl even 12 1
3660.1.dh.d yes 4 12.b even 2 1
3660.1.dh.d yes 4 20.d odd 2 1
3660.1.dh.d yes 4 183.o even 12 1
3660.1.dh.d yes 4 305.w odd 12 1

Hecke kernels

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

\( T_{7} \) Copy content Toggle raw display
\( T_{17}^{4} - 2T_{17}^{3} + 5T_{17}^{2} - 4T_{17} + 1 \) Copy content Toggle raw display
\( T_{23}^{4} + 2T_{23}^{3} + 2T_{23}^{2} - 2T_{23} + 1 \) Copy content Toggle raw display
\( T_{29} \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - T^{2} + 1 \) Copy content Toggle raw display
$3$ \( (T + 1)^{4} \) Copy content Toggle raw display
$5$ \( T^{4} - T^{2} + 1 \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( T^{4} \) Copy content Toggle raw display
$13$ \( T^{4} \) Copy content Toggle raw display
$17$ \( T^{4} - 2 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$19$ \( T^{4} + 3T^{2} + 9 \) Copy content Toggle raw display
$23$ \( T^{4} + 2 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$29$ \( T^{4} \) Copy content Toggle raw display
$31$ \( T^{4} - 4 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$37$ \( T^{4} \) Copy content Toggle raw display
$41$ \( T^{4} \) Copy content Toggle raw display
$43$ \( T^{4} \) Copy content Toggle raw display
$47$ \( T^{4} - T^{2} + 1 \) Copy content Toggle raw display
$53$ \( T^{4} - 2 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$59$ \( T^{4} \) Copy content Toggle raw display
$61$ \( T^{4} - T^{2} + 1 \) Copy content Toggle raw display
$67$ \( T^{4} \) Copy content Toggle raw display
$71$ \( T^{4} \) Copy content Toggle raw display
$73$ \( T^{4} \) Copy content Toggle raw display
$79$ \( T^{4} + 2 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$83$ \( (T^{2} - 3 T + 3)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} \) Copy content Toggle raw display
$97$ \( T^{4} \) Copy content Toggle raw display
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