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Properties

Label 3672.1.dp.b
Level $3672$
Weight $1$
Character orbit 3672.dp
Analytic conductor $1.833$
Analytic rank $0$
Dimension $12$
Projective image $D_{36}$
CM discriminant -8
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3672,1,Mod(115,3672)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3672, base_ring=CyclotomicField(36))
 
chi = DirichletCharacter(H, H._module([18, 18, 32, 9]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3672.115");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3672 = 2^{3} \cdot 3^{3} \cdot 17 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3672.dp (of order \(36\), degree \(12\), 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: \(1.83256672639\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\Q(\zeta_{36})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - x^{6} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{36}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{36} - \cdots)\)

$q$-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.

\(f(q)\) \(=\) \( q - \zeta_{36}^{7} q^{2} + \zeta_{36} q^{3} + \zeta_{36}^{14} q^{4} - \zeta_{36}^{8} q^{6} + \zeta_{36}^{3} q^{8} + \zeta_{36}^{2} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - \zeta_{36}^{7} q^{2} + \zeta_{36} q^{3} + \zeta_{36}^{14} q^{4} - \zeta_{36}^{8} q^{6} + \zeta_{36}^{3} q^{8} + \zeta_{36}^{2} q^{9} + ( - \zeta_{36}^{15} - \zeta_{36}^{14}) q^{11} + \zeta_{36}^{15} q^{12} - \zeta_{36}^{10} q^{16} + \zeta_{36}^{17} q^{17} - \zeta_{36}^{9} q^{18} + ( - \zeta_{36}^{13} + \zeta_{36}^{11}) q^{19} + ( - \zeta_{36}^{4} - \zeta_{36}^{3}) q^{22} + \zeta_{36}^{4} q^{24} + \zeta_{36}^{7} q^{25} + \zeta_{36}^{3} q^{27} + \zeta_{36}^{17} q^{32} + ( - \zeta_{36}^{16} - \zeta_{36}^{15}) q^{33} + \zeta_{36}^{6} q^{34} + \zeta_{36}^{16} q^{36} + ( - \zeta_{36}^{2} + 1) q^{38} + ( - \zeta_{36}^{12} - \zeta_{36}) q^{41} + (\zeta_{36}^{12} - \zeta_{36}^{8}) q^{43} + (\zeta_{36}^{11} + \zeta_{36}^{10}) q^{44} - \zeta_{36}^{11} q^{48} + \zeta_{36}^{17} q^{49} - \zeta_{36}^{14} q^{50} - q^{51} - \zeta_{36}^{10} q^{54} + ( - \zeta_{36}^{14} + \zeta_{36}^{12}) q^{57} + ( - \zeta_{36}^{16} + 1) q^{59} + \zeta_{36}^{6} q^{64} + ( - \zeta_{36}^{5} - \zeta_{36}^{4}) q^{66} + (\zeta_{36}^{15} - \zeta_{36}^{7}) q^{67} - \zeta_{36}^{13} q^{68} + \zeta_{36}^{5} q^{72} + (\zeta_{36}^{10} + \zeta_{36}^{5}) q^{73} + \zeta_{36}^{8} q^{75} + (\zeta_{36}^{9} - \zeta_{36}^{7}) q^{76} + \zeta_{36}^{4} q^{81} + (\zeta_{36}^{8} - \zeta_{36}) q^{82} + (\zeta_{36}^{10} + \zeta_{36}^{4}) q^{83} + (\zeta_{36}^{15} + \zeta_{36}) q^{86} + ( - \zeta_{36}^{17} + 1) q^{88} + ( - \zeta_{36}^{15} - \zeta_{36}^{9}) q^{89} - q^{96} + ( - \zeta_{36}^{6} - \zeta_{36}^{5}) q^{97} + \zeta_{36}^{6} q^{98} + ( - \zeta_{36}^{17} - \zeta_{36}^{16}) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 6 q^{34} + 12 q^{38} + 6 q^{41} - 6 q^{43} - 12 q^{51} - 6 q^{57} + 12 q^{59} + 6 q^{64} + 12 q^{88} - 12 q^{96} - 6 q^{97} + 6 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3672\mathbb{Z}\right)^\times\).

\(n\) \(137\) \(649\) \(919\) \(1837\)
\(\chi(n)\) \(-\zeta_{36}^{14}\) \(-\zeta_{36}^{9}\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
115.1
−0.984808 − 0.173648i
−0.342020 − 0.939693i
−0.642788 − 0.766044i
0.342020 − 0.939693i
0.342020 + 0.939693i
−0.642788 + 0.766044i
−0.342020 + 0.939693i
−0.984808 + 0.173648i
0.642788 − 0.766044i
0.984808 + 0.173648i
0.984808 − 0.173648i
0.642788 + 0.766044i
0.342020 + 0.939693i −0.984808 − 0.173648i −0.766044 + 0.642788i 0 −0.173648 − 0.984808i 0 −0.866025 − 0.500000i 0.939693 + 0.342020i 0
259.1 −0.642788 + 0.766044i −0.342020 − 0.939693i −0.173648 − 0.984808i 0 0.939693 + 0.342020i 0 0.866025 + 0.500000i −0.766044 + 0.642788i 0
931.1 0.984808 − 0.173648i −0.642788 − 0.766044i 0.939693 − 0.342020i 0 −0.766044 − 0.642788i 0 0.866025 − 0.500000i −0.173648 + 0.984808i 0
1075.1 0.642788 + 0.766044i 0.342020 − 0.939693i −0.173648 + 0.984808i 0 0.939693 − 0.342020i 0 −0.866025 + 0.500000i −0.766044 − 0.642788i 0
1339.1 0.642788 − 0.766044i 0.342020 + 0.939693i −0.173648 − 0.984808i 0 0.939693 + 0.342020i 0 −0.866025 − 0.500000i −0.766044 + 0.642788i 0
1483.1 0.984808 + 0.173648i −0.642788 + 0.766044i 0.939693 + 0.342020i 0 −0.766044 + 0.642788i 0 0.866025 + 0.500000i −0.173648 − 0.984808i 0
2155.1 −0.642788 − 0.766044i −0.342020 + 0.939693i −0.173648 + 0.984808i 0 0.939693 − 0.342020i 0 0.866025 − 0.500000i −0.766044 − 0.642788i 0
2299.1 0.342020 − 0.939693i −0.984808 + 0.173648i −0.766044 − 0.642788i 0 −0.173648 + 0.984808i 0 −0.866025 + 0.500000i 0.939693 − 0.342020i 0
2563.1 −0.984808 − 0.173648i 0.642788 − 0.766044i 0.939693 + 0.342020i 0 −0.766044 + 0.642788i 0 −0.866025 − 0.500000i −0.173648 − 0.984808i 0
2707.1 −0.342020 − 0.939693i 0.984808 + 0.173648i −0.766044 + 0.642788i 0 −0.173648 − 0.984808i 0 0.866025 + 0.500000i 0.939693 + 0.342020i 0
3379.1 −0.342020 + 0.939693i 0.984808 − 0.173648i −0.766044 − 0.642788i 0 −0.173648 + 0.984808i 0 0.866025 − 0.500000i 0.939693 − 0.342020i 0
3523.1 −0.984808 + 0.173648i 0.642788 + 0.766044i 0.939693 − 0.342020i 0 −0.766044 − 0.642788i 0 −0.866025 + 0.500000i −0.173648 + 0.984808i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 115.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 CM by \(\Q(\sqrt{-2}) \)
459.x even 36 1 inner
3672.dp odd 36 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3672.1.dp.b yes 12
8.d odd 2 1 CM 3672.1.dp.b yes 12
17.c even 4 1 3672.1.dp.a ✓ 12
27.e even 9 1 3672.1.dp.a ✓ 12
136.j odd 4 1 3672.1.dp.a ✓ 12
216.r odd 18 1 3672.1.dp.a ✓ 12
459.x even 36 1 inner 3672.1.dp.b yes 12
3672.dp odd 36 1 inner 3672.1.dp.b yes 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3672.1.dp.a ✓ 12 17.c even 4 1
3672.1.dp.a ✓ 12 27.e even 9 1
3672.1.dp.a ✓ 12 136.j odd 4 1
3672.1.dp.a ✓ 12 216.r odd 18 1
3672.1.dp.b yes 12 1.a even 1 1 trivial
3672.1.dp.b yes 12 8.d odd 2 1 CM
3672.1.dp.b yes 12 459.x even 36 1 inner
3672.1.dp.b yes 12 3672.dp odd 36 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{11}^{12} - 3 T_{11}^{10} + 2 T_{11}^{9} + 6 T_{11}^{8} + 18 T_{11}^{7} - 4 T_{11}^{6} - 30 T_{11}^{5} + \cdots + 1 \) acting on \(S_{1}^{\mathrm{new}}(3672, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} - T^{6} + 1 \) Copy content Toggle raw display
$3$ \( T^{12} - T^{6} + 1 \) Copy content Toggle raw display
$5$ \( T^{12} \) Copy content Toggle raw display
$7$ \( T^{12} \) Copy content Toggle raw display
$11$ \( T^{12} - 3 T^{10} + \cdots + 1 \) Copy content Toggle raw display
$13$ \( T^{12} \) Copy content Toggle raw display
$17$ \( T^{12} - T^{6} + 1 \) Copy content Toggle raw display
$19$ \( T^{12} - 6 T^{10} + \cdots + 1 \) Copy content Toggle raw display
$23$ \( T^{12} \) Copy content Toggle raw display
$29$ \( T^{12} \) Copy content Toggle raw display
$31$ \( T^{12} \) Copy content Toggle raw display
$37$ \( T^{12} \) Copy content Toggle raw display
$41$ \( T^{12} - 6 T^{11} + \cdots + 1 \) Copy content Toggle raw display
$43$ \( (T^{6} + 3 T^{5} + 6 T^{4} + \cdots + 3)^{2} \) Copy content Toggle raw display
$47$ \( T^{12} \) Copy content Toggle raw display
$53$ \( T^{12} \) Copy content Toggle raw display
$59$ \( (T^{6} - 6 T^{5} + 15 T^{4} + \cdots + 3)^{2} \) Copy content Toggle raw display
$61$ \( T^{12} \) Copy content Toggle raw display
$67$ \( T^{12} - 3 T^{10} + \cdots + 9 \) Copy content Toggle raw display
$71$ \( T^{12} \) Copy content Toggle raw display
$73$ \( T^{12} - 2 T^{9} + \cdots + 1 \) Copy content Toggle raw display
$79$ \( T^{12} \) Copy content Toggle raw display
$83$ \( (T^{6} + 9 T^{3} + 27)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} + 3 T^{2} + 9)^{3} \) Copy content Toggle raw display
$97$ \( T^{12} + 6 T^{11} + \cdots + 1 \) Copy content Toggle raw display
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