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Properties

Label 984.1.cj.b
Level $984$
Weight $1$
Character orbit 984.cj
Analytic conductor $0.491$
Analytic rank $0$
Dimension $16$
Projective image $D_{40}$
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] = [984,1,Mod(11,984)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(984, base_ring=CyclotomicField(40))
 
chi = DirichletCharacter(H, H._module([20, 20, 20, 3]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("984.11");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 984 = 2^{3} \cdot 3 \cdot 41 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 984.cj (of order \(40\), degree \(16\), 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: \(0.491079972431\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\Q(\zeta_{40})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - x^{12} + x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{40}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{40} - \cdots)\)

$q$-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.

\(f(q)\) \(=\) \( q + \zeta_{40}^{13} q^{2} + \zeta_{40}^{17} q^{3} - \zeta_{40}^{6} q^{4} - \zeta_{40}^{10} q^{6} - \zeta_{40}^{19} q^{8} - \zeta_{40}^{14} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{40}^{13} q^{2} + \zeta_{40}^{17} q^{3} - \zeta_{40}^{6} q^{4} - \zeta_{40}^{10} q^{6} - \zeta_{40}^{19} q^{8} - \zeta_{40}^{14} q^{9} + (\zeta_{40}^{18} + \zeta_{40}^{15}) q^{11} + \zeta_{40}^{3} q^{12} + \zeta_{40}^{12} q^{16} + (\zeta_{40}^{19} - \zeta_{40}^{4}) q^{17} + \zeta_{40}^{7} q^{18} + (\zeta_{40}^{10} - \zeta_{40}^{9}) q^{19} + ( - \zeta_{40}^{11} - \zeta_{40}^{8}) q^{22} + \zeta_{40}^{16} q^{24} - \zeta_{40}^{2} q^{25} + \zeta_{40}^{11} q^{27} - \zeta_{40}^{5} q^{32} + ( - \zeta_{40}^{15} - \zeta_{40}^{12}) q^{33} + ( - \zeta_{40}^{17} - \zeta_{40}^{12}) q^{34} - q^{36} + ( - \zeta_{40}^{3} + \zeta_{40}^{2}) q^{38} - \zeta_{40}^{11} q^{41} + ( - \zeta_{40}^{5} - \zeta_{40}) q^{43} + (\zeta_{40}^{4} + \zeta_{40}) q^{44} - \zeta_{40}^{9} q^{48} + \zeta_{40}^{9} q^{49} - \zeta_{40}^{15} q^{50} + ( - \zeta_{40}^{16} + \zeta_{40}) q^{51} - \zeta_{40}^{4} q^{54} + ( - \zeta_{40}^{7} + \zeta_{40}^{6}) q^{57} + ( - \zeta_{40}^{13} + \zeta_{40}^{3}) q^{59} - \zeta_{40}^{18} q^{64} + (\zeta_{40}^{8} + \zeta_{40}^{5}) q^{66} + ( - \zeta_{40}^{7} - 1) q^{67} + (\zeta_{40}^{10} + \zeta_{40}^{5}) q^{68} - \zeta_{40}^{13} q^{72} + (\zeta_{40}^{7} - \zeta_{40}^{3}) q^{73} - \zeta_{40}^{19} q^{75} + ( - \zeta_{40}^{16} + \zeta_{40}^{15}) q^{76} - \zeta_{40}^{8} q^{81} + \zeta_{40}^{4} q^{82} + (\zeta_{40}^{14} + \zeta_{40}^{6}) q^{83} + ( - \zeta_{40}^{18} - \zeta_{40}^{14}) q^{86} + (\zeta_{40}^{17} + \zeta_{40}^{14}) q^{88} + (\zeta_{40}^{8} - \zeta_{40}) q^{89} + \zeta_{40}^{2} q^{96} + (\zeta_{40}^{14} + \zeta_{40}^{13}) q^{97} - \zeta_{40}^{2} q^{98} + (\zeta_{40}^{12} + \zeta_{40}^{9}) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 4 q^{16} - 4 q^{17} + 4 q^{22} - 4 q^{24} - 4 q^{33} - 4 q^{34} - 16 q^{36} + 4 q^{44} + 4 q^{51} - 4 q^{54} - 4 q^{66} - 16 q^{67} + 4 q^{76} + 4 q^{81} + 4 q^{82} - 4 q^{89} + 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/984\mathbb{Z}\right)^\times\).

\(n\) \(247\) \(329\) \(457\) \(493\)
\(\chi(n)\) \(-1\) \(-1\) \(-\zeta_{40}^{11}\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
11.1
−0.453990 + 0.891007i
−0.156434 + 0.987688i
−0.453990 − 0.891007i
−0.987688 + 0.156434i
−0.891007 + 0.453990i
0.891007 − 0.453990i
0.987688 − 0.156434i
0.453990 + 0.891007i
0.156434 − 0.987688i
0.453990 − 0.891007i
0.987688 + 0.156434i
−0.156434 − 0.987688i
−0.891007 − 0.453990i
0.891007 + 0.453990i
0.156434 + 0.987688i
−0.987688 − 0.156434i
0.156434 + 0.987688i −0.987688 − 0.156434i −0.951057 + 0.309017i 0 − 1.00000i 0 −0.453990 − 0.891007i 0.951057 + 0.309017i 0
35.1 −0.891007 − 0.453990i −0.453990 − 0.891007i 0.587785 + 0.809017i 0 1.00000i 0 −0.156434 − 0.987688i −0.587785 + 0.809017i 0
179.1 0.156434 − 0.987688i −0.987688 + 0.156434i −0.951057 − 0.309017i 0 1.00000i 0 −0.453990 + 0.891007i 0.951057 − 0.309017i 0
227.1 0.453990 + 0.891007i 0.891007 + 0.453990i −0.587785 + 0.809017i 0 1.00000i 0 −0.987688 − 0.156434i 0.587785 + 0.809017i 0
275.1 −0.987688 − 0.156434i 0.156434 + 0.987688i 0.951057 + 0.309017i 0 − 1.00000i 0 −0.891007 − 0.453990i −0.951057 + 0.309017i 0
299.1 0.987688 + 0.156434i −0.156434 − 0.987688i 0.951057 + 0.309017i 0 − 1.00000i 0 0.891007 + 0.453990i −0.951057 + 0.309017i 0
347.1 −0.453990 − 0.891007i −0.891007 − 0.453990i −0.587785 + 0.809017i 0 1.00000i 0 0.987688 + 0.156434i 0.587785 + 0.809017i 0
395.1 −0.156434 + 0.987688i 0.987688 − 0.156434i −0.951057 − 0.309017i 0 1.00000i 0 0.453990 − 0.891007i 0.951057 − 0.309017i 0
539.1 0.891007 + 0.453990i 0.453990 + 0.891007i 0.587785 + 0.809017i 0 1.00000i 0 0.156434 + 0.987688i −0.587785 + 0.809017i 0
563.1 −0.156434 − 0.987688i 0.987688 + 0.156434i −0.951057 + 0.309017i 0 − 1.00000i 0 0.453990 + 0.891007i 0.951057 + 0.309017i 0
587.1 −0.453990 + 0.891007i −0.891007 + 0.453990i −0.587785 − 0.809017i 0 − 1.00000i 0 0.987688 − 0.156434i 0.587785 − 0.809017i 0
731.1 −0.891007 + 0.453990i −0.453990 + 0.891007i 0.587785 − 0.809017i 0 − 1.00000i 0 −0.156434 + 0.987688i −0.587785 − 0.809017i 0
755.1 −0.987688 + 0.156434i 0.156434 − 0.987688i 0.951057 − 0.309017i 0 1.00000i 0 −0.891007 + 0.453990i −0.951057 − 0.309017i 0
803.1 0.987688 − 0.156434i −0.156434 + 0.987688i 0.951057 − 0.309017i 0 1.00000i 0 0.891007 − 0.453990i −0.951057 − 0.309017i 0
827.1 0.891007 − 0.453990i 0.453990 − 0.891007i 0.587785 − 0.809017i 0 − 1.00000i 0 0.156434 − 0.987688i −0.587785 − 0.809017i 0
971.1 0.453990 − 0.891007i 0.891007 − 0.453990i −0.587785 − 0.809017i 0 − 1.00000i 0 −0.987688 + 0.156434i 0.587785 − 0.809017i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 11.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 CM by \(\Q(\sqrt{-2}) \)
123.o even 40 1 inner
984.cj odd 40 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 984.1.cj.b yes 16
3.b odd 2 1 984.1.cj.a ✓ 16
4.b odd 2 1 3936.1.fx.a 16
8.b even 2 1 3936.1.fx.a 16
8.d odd 2 1 CM 984.1.cj.b yes 16
12.b even 2 1 3936.1.fx.b 16
24.f even 2 1 984.1.cj.a ✓ 16
24.h odd 2 1 3936.1.fx.b 16
41.h odd 40 1 984.1.cj.a ✓ 16
123.o even 40 1 inner 984.1.cj.b yes 16
164.o even 40 1 3936.1.fx.b 16
328.bd even 40 1 984.1.cj.a ✓ 16
328.bf odd 40 1 3936.1.fx.b 16
492.be odd 40 1 3936.1.fx.a 16
984.ch even 40 1 3936.1.fx.a 16
984.cj odd 40 1 inner 984.1.cj.b yes 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
984.1.cj.a ✓ 16 3.b odd 2 1
984.1.cj.a ✓ 16 24.f even 2 1
984.1.cj.a ✓ 16 41.h odd 40 1
984.1.cj.a ✓ 16 328.bd even 40 1
984.1.cj.b yes 16 1.a even 1 1 trivial
984.1.cj.b yes 16 8.d odd 2 1 CM
984.1.cj.b yes 16 123.o even 40 1 inner
984.1.cj.b yes 16 984.cj odd 40 1 inner
3936.1.fx.a 16 4.b odd 2 1
3936.1.fx.a 16 8.b even 2 1
3936.1.fx.a 16 492.be odd 40 1
3936.1.fx.a 16 984.ch even 40 1
3936.1.fx.b 16 12.b even 2 1
3936.1.fx.b 16 24.h odd 2 1
3936.1.fx.b 16 164.o even 40 1
3936.1.fx.b 16 328.bf odd 40 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{11}^{16} - 2 T_{11}^{14} + 4 T_{11}^{13} + 7 T_{11}^{12} - 16 T_{11}^{11} + 6 T_{11}^{10} - 8 T_{11}^{9} + \cdots + 1 \) acting on \(S_{1}^{\mathrm{new}}(984, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} - T^{12} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( T^{16} - T^{12} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( T^{16} \) Copy content Toggle raw display
$7$ \( T^{16} \) Copy content Toggle raw display
$11$ \( T^{16} - 2 T^{14} + \cdots + 1 \) Copy content Toggle raw display
$13$ \( T^{16} \) Copy content Toggle raw display
$17$ \( T^{16} + 4 T^{15} + \cdots + 16 \) Copy content Toggle raw display
$19$ \( T^{16} + 8 T^{14} + \cdots + 1 \) Copy content Toggle raw display
$23$ \( T^{16} \) Copy content Toggle raw display
$29$ \( T^{16} \) Copy content Toggle raw display
$31$ \( T^{16} \) Copy content Toggle raw display
$37$ \( T^{16} \) Copy content Toggle raw display
$41$ \( T^{16} - T^{12} + \cdots + 1 \) Copy content Toggle raw display
$43$ \( T^{16} + 5 T^{12} + \cdots + 625 \) Copy content Toggle raw display
$47$ \( T^{16} \) Copy content Toggle raw display
$53$ \( T^{16} \) Copy content Toggle raw display
$59$ \( (T^{8} - 2 T^{6} + 4 T^{4} + \cdots + 16)^{2} \) Copy content Toggle raw display
$61$ \( T^{16} \) Copy content Toggle raw display
$67$ \( T^{16} + 16 T^{15} + \cdots + 1 \) Copy content Toggle raw display
$71$ \( T^{16} \) Copy content Toggle raw display
$73$ \( (T^{8} + 7 T^{4} + 1)^{2} \) Copy content Toggle raw display
$79$ \( T^{16} \) Copy content Toggle raw display
$83$ \( (T^{4} + 3 T^{2} + 1)^{4} \) Copy content Toggle raw display
$89$ \( T^{16} + 4 T^{15} + \cdots + 1 \) Copy content Toggle raw display
$97$ \( T^{16} - 2 T^{14} + \cdots + 1 \) Copy content Toggle raw display
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