Properties

Label 3672.1.df.b
Level $3672$
Weight $1$
Character orbit 3672.df
Analytic conductor $1.833$
Analytic rank $0$
Dimension $8$
Projective image $D_{24}$
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(19,3672)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3672, base_ring=CyclotomicField(24))
 
chi = DirichletCharacter(H, H._module([12, 12, 16, 21]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3672.19");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3672 = 2^{3} \cdot 3^{3} \cdot 17 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3672.df (of order \(24\), degree \(8\), 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: \(1.83256672639\)
Analytic rank: \(0\)
Dimension: \(8\)
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: no (minimal twist has level 1224)
Projective image: \(D_{24}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{24} - \cdots)\)

$q$-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.

\(f(q)\) \(=\) \( q - \zeta_{24}^{5} q^{2} + \zeta_{24}^{10} q^{4} + \zeta_{24}^{3} q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q - \zeta_{24}^{5} q^{2} + \zeta_{24}^{10} q^{4} + \zeta_{24}^{3} q^{8} + (\zeta_{24} + 1) q^{11} - \zeta_{24}^{8} q^{16} + \zeta_{24}^{5} q^{17} + (\zeta_{24}^{11} + \zeta_{24}^{7}) q^{19} + ( - \zeta_{24}^{6} - \zeta_{24}^{5}) q^{22} + \zeta_{24}^{11} q^{25} - \zeta_{24} q^{32} - \zeta_{24}^{10} q^{34} + (\zeta_{24}^{4} + 1) q^{38} + (\zeta_{24}^{3} + \zeta_{24}^{2}) q^{41} + (\zeta_{24}^{6} - \zeta_{24}^{4}) q^{43} + (\zeta_{24}^{11} + \zeta_{24}^{10}) q^{44} - \zeta_{24} q^{49} + \zeta_{24}^{4} q^{50} + (\zeta_{24}^{8} + \zeta_{24}^{6}) q^{59} + \zeta_{24}^{6} q^{64} + ( - \zeta_{24}^{11} + \zeta_{24}^{9}) q^{67} - \zeta_{24}^{3} q^{68} + ( - \zeta_{24}^{10} + \zeta_{24}^{5}) q^{73} + ( - \zeta_{24}^{9} - \zeta_{24}^{5}) q^{76} + ( - \zeta_{24}^{8} - \zeta_{24}^{7}) q^{82} + ( - \zeta_{24}^{8} - \zeta_{24}^{2}) q^{83} + ( - \zeta_{24}^{11} + \zeta_{24}^{9}) q^{86} + (\zeta_{24}^{4} + \zeta_{24}^{3}) q^{88} + ( - \zeta_{24}^{9} - \zeta_{24}^{3}) q^{89} + (\zeta_{24}^{4} - \zeta_{24}^{3}) q^{97} + \zeta_{24}^{6} q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 8 q^{11} + 4 q^{16} + 12 q^{38} - 4 q^{43} + 4 q^{50} - 4 q^{59} + 4 q^{82} + 4 q^{83} + 4 q^{88} + 4 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(137\) \(649\) \(919\) \(1837\)
\(\chi(n)\) \(-\zeta_{24}^{4}\) \(-\zeta_{24}^{3}\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1
0.258819 0.965926i
−0.965926 0.258819i
−0.258819 0.965926i
0.965926 0.258819i
0.965926 + 0.258819i
−0.258819 + 0.965926i
−0.965926 + 0.258819i
0.258819 + 0.965926i
−0.965926 + 0.258819i 0 0.866025 0.500000i 0 0 0 −0.707107 + 0.707107i 0 0
451.1 0.258819 + 0.965926i 0 −0.866025 + 0.500000i 0 0 0 −0.707107 0.707107i 0 0
739.1 0.965926 + 0.258819i 0 0.866025 + 0.500000i 0 0 0 0.707107 + 0.707107i 0 0
1171.1 −0.258819 + 0.965926i 0 −0.866025 0.500000i 0 0 0 0.707107 0.707107i 0 0
1963.1 −0.258819 0.965926i 0 −0.866025 + 0.500000i 0 0 0 0.707107 + 0.707107i 0 0
2395.1 0.965926 0.258819i 0 0.866025 0.500000i 0 0 0 0.707107 0.707107i 0 0
2467.1 0.258819 0.965926i 0 −0.866025 0.500000i 0 0 0 −0.707107 + 0.707107i 0 0
2899.1 −0.965926 0.258819i 0 0.866025 + 0.500000i 0 0 0 −0.707107 0.707107i 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 19.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 CM by \(\Q(\sqrt{-2}) \)
153.r even 24 1 inner
1224.cp odd 24 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3672.1.df.b 8
3.b odd 2 1 1224.1.cp.b yes 8
8.d odd 2 1 CM 3672.1.df.b 8
9.c even 3 1 3672.1.df.a 8
9.d odd 6 1 1224.1.cp.a 8
17.d even 8 1 3672.1.df.a 8
24.f even 2 1 1224.1.cp.b yes 8
51.g odd 8 1 1224.1.cp.a 8
72.l even 6 1 1224.1.cp.a 8
72.p odd 6 1 3672.1.df.a 8
136.p odd 8 1 3672.1.df.a 8
153.q odd 24 1 1224.1.cp.b yes 8
153.r even 24 1 inner 3672.1.df.b 8
408.bd even 8 1 1224.1.cp.a 8
1224.cn even 24 1 1224.1.cp.b yes 8
1224.cp odd 24 1 inner 3672.1.df.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1224.1.cp.a 8 9.d odd 6 1
1224.1.cp.a 8 51.g odd 8 1
1224.1.cp.a 8 72.l even 6 1
1224.1.cp.a 8 408.bd even 8 1
1224.1.cp.b yes 8 3.b odd 2 1
1224.1.cp.b yes 8 24.f even 2 1
1224.1.cp.b yes 8 153.q odd 24 1
1224.1.cp.b yes 8 1224.cn even 24 1
3672.1.df.a 8 9.c even 3 1
3672.1.df.a 8 17.d even 8 1
3672.1.df.a 8 72.p odd 6 1
3672.1.df.a 8 136.p odd 8 1
3672.1.df.b 8 1.a even 1 1 trivial
3672.1.df.b 8 8.d odd 2 1 CM
3672.1.df.b 8 153.r even 24 1 inner
3672.1.df.b 8 1224.cp odd 24 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{11}^{8} - 8T_{11}^{7} + 28T_{11}^{6} - 56T_{11}^{5} + 69T_{11}^{4} - 52T_{11}^{3} + 22T_{11}^{2} - 4T_{11} + 1 \) acting on \(S_{1}^{\mathrm{new}}(3672, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} - T^{4} + 1 \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( T^{8} \) Copy content Toggle raw display
$11$ \( T^{8} - 8 T^{7} + 28 T^{6} - 56 T^{5} + \cdots + 1 \) Copy content Toggle raw display
$13$ \( T^{8} \) Copy content Toggle raw display
$17$ \( T^{8} - T^{4} + 1 \) Copy content Toggle raw display
$19$ \( (T^{4} + 9)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} \) Copy content Toggle raw display
$29$ \( T^{8} \) Copy content Toggle raw display
$31$ \( T^{8} \) Copy content Toggle raw display
$37$ \( T^{8} \) Copy content Toggle raw display
$41$ \( T^{8} - 2 T^{6} + 4 T^{5} + 5 T^{4} + \cdots + 1 \) Copy content Toggle raw display
$43$ \( (T^{4} + 2 T^{3} + 5 T^{2} + 4 T + 1)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} \) Copy content Toggle raw display
$53$ \( T^{8} \) Copy content Toggle raw display
$59$ \( (T^{4} + 2 T^{3} + 5 T^{2} + 4 T + 1)^{2} \) Copy content Toggle raw display
$61$ \( T^{8} \) Copy content Toggle raw display
$67$ \( T^{8} + 4 T^{6} + 15 T^{4} + 4 T^{2} + \cdots + 1 \) Copy content Toggle raw display
$71$ \( T^{8} \) Copy content Toggle raw display
$73$ \( T^{8} - 2 T^{6} + 4 T^{5} + 2 T^{4} + \cdots + 1 \) Copy content Toggle raw display
$79$ \( T^{8} \) Copy content Toggle raw display
$83$ \( (T^{4} - 2 T^{3} + 2 T^{2} - 4 T + 4)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 2)^{4} \) Copy content Toggle raw display
$97$ \( T^{8} - 4 T^{7} + 10 T^{6} - 16 T^{5} + \cdots + 1 \) Copy content Toggle raw display
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