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Properties

Label 3660.1.dh.f
Level $3660$
Weight $1$
Character orbit 3660.dh
Analytic conductor $1.827$
Analytic rank $0$
Dimension $8$
Projective image $D_{12}$
CM discriminant -20
Inner twists $8$

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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: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{12})\)
Coefficient field: \(\Q(\zeta_{24})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{4} + 1 \) 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.

\(f(q)\) \(=\) \( q - \zeta_{24}^{5} q^{2} - \zeta_{24}^{7} q^{3} + \zeta_{24}^{10} q^{4} - \zeta_{24}^{8} q^{5} - q^{6} - 2 \zeta_{24}^{11} q^{7} + \zeta_{24}^{3} q^{8} - \zeta_{24}^{2} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - \zeta_{24}^{5} q^{2} - \zeta_{24}^{7} q^{3} + \zeta_{24}^{10} q^{4} - \zeta_{24}^{8} q^{5} - q^{6} - 2 \zeta_{24}^{11} q^{7} + \zeta_{24}^{3} q^{8} - \zeta_{24}^{2} q^{9} - \zeta_{24} q^{10} + \zeta_{24}^{5} q^{12} - 2 \zeta_{24}^{4} q^{14} - \zeta_{24}^{3} q^{15} - \zeta_{24}^{8} q^{16} + \zeta_{24}^{7} q^{18} + \zeta_{24}^{6} q^{20} - 2 \zeta_{24}^{6} q^{21} + (\zeta_{24}^{11} + \zeta_{24}^{7}) q^{23} - \zeta_{24}^{10} q^{24} - \zeta_{24}^{4} q^{25} + \zeta_{24}^{9} q^{27} + 2 \zeta_{24}^{9} q^{28} + ( - \zeta_{24}^{2} + 1) q^{29} + \zeta_{24}^{8} q^{30} - \zeta_{24} q^{32} - 2 \zeta_{24}^{7} q^{35} + q^{36} - \zeta_{24}^{11} q^{40} - q^{41} + 2 \zeta_{24}^{11} q^{42} + \zeta_{24}^{5} q^{43} + \zeta_{24}^{10} q^{45} + (\zeta_{24}^{4} + 1) q^{46} + ( - \zeta_{24}^{7} + \zeta_{24}) q^{47} - \zeta_{24}^{3} q^{48} - 3 \zeta_{24}^{10} q^{49} + \zeta_{24}^{9} q^{50} + \zeta_{24}^{2} q^{54} + 2 \zeta_{24}^{2} q^{56} + (\zeta_{24}^{7} - \zeta_{24}^{5}) q^{58} + \zeta_{24} q^{60} - \zeta_{24}^{6} q^{61} - 2 \zeta_{24} q^{63} + \zeta_{24}^{6} q^{64} - \zeta_{24}^{5} q^{67} + (\zeta_{24}^{6} + \zeta_{24}^{2}) q^{69} - 2 q^{70} - \zeta_{24}^{5} q^{72} + \zeta_{24}^{11} q^{75} - \zeta_{24}^{4} q^{80} + \zeta_{24}^{4} q^{81} + \zeta_{24}^{5} q^{82} + (\zeta_{24}^{11} - \zeta_{24}^{5}) q^{83} + 2 \zeta_{24}^{4} q^{84} - \zeta_{24}^{10} q^{86} + (\zeta_{24}^{9} - \zeta_{24}^{7}) q^{87} + ( - \zeta_{24}^{10} + \zeta_{24}^{8}) q^{89} + \zeta_{24}^{3} q^{90} + ( - \zeta_{24}^{9} - \zeta_{24}^{5}) q^{92} + ( - \zeta_{24}^{6} - 1) q^{94} + \zeta_{24}^{8} q^{96} - 3 \zeta_{24}^{3} q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4 q^{5} - 8 q^{6}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 4 q^{5} - 8 q^{6} - 8 q^{14} + 4 q^{16} - 4 q^{25} + 8 q^{29} - 4 q^{30} + 8 q^{36} - 8 q^{41} + 12 q^{46} - 16 q^{70} - 4 q^{80} + 4 q^{81} + 8 q^{84} - 4 q^{89} - 8 q^{94} - 4 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

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

\(n\) \(1831\) \(2197\) \(2441\) \(3601\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\) \(-\zeta_{24}^{10}\)

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
0.258819 − 0.965926i
−0.258819 + 0.965926i
0.965926 − 0.258819i
−0.965926 + 0.258819i
0.258819 + 0.965926i
−0.258819 − 0.965926i
0.965926 + 0.258819i
−0.965926 − 0.258819i
−0.965926 + 0.258819i 0.965926 + 0.258819i 0.866025 − 0.500000i 0.500000 − 0.866025i −1.00000 0.517638 + 1.93185i −0.707107 + 0.707107i 0.866025 + 0.500000i −0.258819 + 0.965926i
1199.2 0.965926 − 0.258819i −0.965926 − 0.258819i 0.866025 − 0.500000i 0.500000 − 0.866025i −1.00000 −0.517638 − 1.93185i 0.707107 − 0.707107i 0.866025 + 0.500000i 0.258819 − 0.965926i
1679.1 −0.258819 + 0.965926i 0.258819 + 0.965926i −0.866025 − 0.500000i 0.500000 + 0.866025i −1.00000 1.93185 + 0.517638i 0.707107 − 0.707107i −0.866025 + 0.500000i −0.965926 + 0.258819i
1679.2 0.258819 − 0.965926i −0.258819 − 0.965926i −0.866025 − 0.500000i 0.500000 + 0.866025i −1.00000 −1.93185 − 0.517638i −0.707107 + 0.707107i −0.866025 + 0.500000i 0.965926 − 0.258819i
1859.1 −0.965926 − 0.258819i 0.965926 − 0.258819i 0.866025 + 0.500000i 0.500000 + 0.866025i −1.00000 0.517638 − 1.93185i −0.707107 − 0.707107i 0.866025 − 0.500000i −0.258819 − 0.965926i
1859.2 0.965926 + 0.258819i −0.965926 + 0.258819i 0.866025 + 0.500000i 0.500000 + 0.866025i −1.00000 −0.517638 + 1.93185i 0.707107 + 0.707107i 0.866025 − 0.500000i 0.258819 + 0.965926i
2339.1 −0.258819 − 0.965926i 0.258819 − 0.965926i −0.866025 + 0.500000i 0.500000 − 0.866025i −1.00000 1.93185 − 0.517638i 0.707107 + 0.707107i −0.866025 − 0.500000i −0.965926 − 0.258819i
2339.2 0.258819 + 0.965926i −0.258819 + 0.965926i −0.866025 + 0.500000i 0.500000 − 0.866025i −1.00000 −1.93185 + 0.517638i −0.707107 − 0.707107i −0.866025 − 0.500000i 0.965926 + 0.258819i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1199.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
20.d odd 2 1 CM by \(\Q(\sqrt{-5}) \)
4.b odd 2 1 inner
5.b even 2 1 inner
183.o even 12 1 inner
732.bc odd 12 1 inner
915.bq 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.f yes 8
3.b odd 2 1 3660.1.dh.e ✓ 8
4.b odd 2 1 inner 3660.1.dh.f yes 8
5.b even 2 1 inner 3660.1.dh.f yes 8
12.b even 2 1 3660.1.dh.e ✓ 8
15.d odd 2 1 3660.1.dh.e ✓ 8
20.d odd 2 1 CM 3660.1.dh.f yes 8
60.h even 2 1 3660.1.dh.e ✓ 8
61.h odd 12 1 3660.1.dh.e ✓ 8
183.o even 12 1 inner 3660.1.dh.f yes 8
244.p even 12 1 3660.1.dh.e ✓ 8
305.w odd 12 1 3660.1.dh.e ✓ 8
732.bc odd 12 1 inner 3660.1.dh.f yes 8
915.bq even 12 1 inner 3660.1.dh.f yes 8
1220.bl even 12 1 3660.1.dh.e ✓ 8
3660.dh odd 12 1 inner 3660.1.dh.f yes 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3660.1.dh.e ✓ 8 3.b odd 2 1
3660.1.dh.e ✓ 8 12.b even 2 1
3660.1.dh.e ✓ 8 15.d odd 2 1
3660.1.dh.e ✓ 8 60.h even 2 1
3660.1.dh.e ✓ 8 61.h odd 12 1
3660.1.dh.e ✓ 8 244.p even 12 1
3660.1.dh.e ✓ 8 305.w odd 12 1
3660.1.dh.e ✓ 8 1220.bl even 12 1
3660.1.dh.f yes 8 1.a even 1 1 trivial
3660.1.dh.f yes 8 4.b odd 2 1 inner
3660.1.dh.f yes 8 5.b even 2 1 inner
3660.1.dh.f yes 8 20.d odd 2 1 CM
3660.1.dh.f yes 8 183.o even 12 1 inner
3660.1.dh.f yes 8 732.bc odd 12 1 inner
3660.1.dh.f yes 8 915.bq even 12 1 inner
3660.1.dh.f yes 8 3660.dh odd 12 1 inner

Hecke kernels

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

\( T_{7}^{8} - 16T_{7}^{4} + 256 \) Copy content Toggle raw display
\( T_{17} \) Copy content Toggle raw display
\( T_{23}^{4} + 9 \) Copy content Toggle raw display
\( T_{29}^{4} - 4T_{29}^{3} + 5T_{29}^{2} - 2T_{29} + 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} - T^{4} + 1 \) Copy content Toggle raw display
$3$ \( T^{8} - T^{4} + 1 \) Copy content Toggle raw display
$5$ \( (T^{2} - T + 1)^{4} \) Copy content Toggle raw display
$7$ \( T^{8} - 16T^{4} + 256 \) Copy content Toggle raw display
$11$ \( T^{8} \) Copy content Toggle raw display
$13$ \( T^{8} \) Copy content Toggle raw display
$17$ \( T^{8} \) Copy content Toggle raw display
$19$ \( T^{8} \) Copy content Toggle raw display
$23$ \( (T^{4} + 9)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} - 4 T^{3} + 5 T^{2} + \cdots + 1)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} \) Copy content Toggle raw display
$37$ \( T^{8} \) Copy content Toggle raw display
$41$ \( (T + 1)^{8} \) Copy content Toggle raw display
$43$ \( T^{8} - T^{4} + 1 \) Copy content Toggle raw display
$47$ \( (T^{4} - 2 T^{2} + 4)^{2} \) Copy content Toggle raw display
$53$ \( T^{8} \) Copy content Toggle raw display
$59$ \( T^{8} \) Copy content Toggle raw display
$61$ \( (T^{2} + 1)^{4} \) Copy content Toggle raw display
$67$ \( T^{8} - T^{4} + 1 \) Copy content Toggle raw display
$71$ \( T^{8} \) Copy content Toggle raw display
$73$ \( T^{8} \) Copy content Toggle raw display
$79$ \( T^{8} \) Copy content Toggle raw display
$83$ \( (T^{4} - 2 T^{2} + 4)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} + 2 T^{3} + 2 T^{2} + \cdots + 1)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} \) Copy content Toggle raw display
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