Properties

Label 1001.1.r.a
Level $1001$
Weight $1$
Character orbit 1001.r
Analytic conductor $0.500$
Analytic rank $0$
Dimension $4$
Projective image $A_{4}$
CM/RM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1001,1,Mod(802,1001)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1001, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([4, 3, 4]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1001.802");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1001 = 7 \cdot 11 \cdot 13 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1001.r (of order \(6\), degree \(2\), 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: \(0.499564077646\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
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: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(A_{4}\)
Projective field: Galois closure of 4.0.1002001.1

$q$-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.

\(f(q)\) \(=\) \( q - \zeta_{12}^{3} q^{2} + \zeta_{12}^{2} q^{5} - \zeta_{12} q^{7} - \zeta_{12}^{3} q^{8} - \zeta_{12}^{4} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - \zeta_{12}^{3} q^{2} + \zeta_{12}^{2} q^{5} - \zeta_{12} q^{7} - \zeta_{12}^{3} q^{8} - \zeta_{12}^{4} q^{9} - \zeta_{12}^{5} q^{10} - \zeta_{12}^{5} q^{11} + \zeta_{12}^{3} q^{13} + \zeta_{12}^{4} q^{14} - q^{16} + \zeta_{12}^{3} q^{17} - \zeta_{12} q^{18} - \zeta_{12}^{2} q^{22} - q^{23} + q^{26} + \zeta_{12} q^{29} - \zeta_{12}^{4} q^{31} + q^{34} - \zeta_{12}^{3} q^{35} - q^{37} - \zeta_{12}^{5} q^{40} - \zeta_{12} q^{41} - \zeta_{12}^{5} q^{43} + q^{45} + \zeta_{12}^{3} q^{46} - \zeta_{12}^{2} q^{47} + \zeta_{12}^{2} q^{49} + \zeta_{12}^{4} q^{53} + \zeta_{12} q^{55} + \zeta_{12}^{4} q^{56} - \zeta_{12}^{4} q^{58} - q^{59} - \zeta_{12} q^{62} + \zeta_{12}^{5} q^{63} - q^{64} + \zeta_{12}^{5} q^{65} + \zeta_{12}^{2} q^{67} - q^{70} + \zeta_{12}^{2} q^{71} - \zeta_{12} q^{72} + \zeta_{12} q^{73} + \zeta_{12}^{3} q^{74} - q^{77} + \zeta_{12}^{5} q^{79} - \zeta_{12}^{2} q^{80} - \zeta_{12}^{2} q^{81} + \zeta_{12}^{4} q^{82} + \zeta_{12}^{5} q^{85} - \zeta_{12}^{2} q^{86} - \zeta_{12}^{2} q^{88} - q^{89} - \zeta_{12}^{3} q^{90} - \zeta_{12}^{4} q^{91} + \zeta_{12}^{5} q^{94} - \zeta_{12}^{2} q^{97} - \zeta_{12}^{5} q^{98} - \zeta_{12}^{3} q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{5} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{5} + 2 q^{9} - 2 q^{14} - 4 q^{16} - 2 q^{22} - 4 q^{23} + 4 q^{26} + 2 q^{31} + 4 q^{34} - 4 q^{37} + 4 q^{45} - 2 q^{47} + 2 q^{49} - 2 q^{53} - 2 q^{56} + 2 q^{58} - 4 q^{59} - 4 q^{64} + 4 q^{67} - 4 q^{70} + 2 q^{71} - 4 q^{77} - 2 q^{80} - 2 q^{81} - 2 q^{82} - 2 q^{86} - 2 q^{88} - 4 q^{89} + 2 q^{91} - 2 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1001\mathbb{Z}\right)^\times\).

\(n\) \(365\) \(430\) \(925\)
\(\chi(n)\) \(-1\) \(-\zeta_{12}^{2}\) \(-\zeta_{12}^{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
802.1
0.866025 + 0.500000i
−0.866025 0.500000i
−0.866025 + 0.500000i
0.866025 0.500000i
1.00000i 0 0 0.500000 + 0.866025i 0 −0.866025 0.500000i 1.00000i 0.500000 0.866025i 0.866025 0.500000i
802.2 1.00000i 0 0 0.500000 + 0.866025i 0 0.866025 + 0.500000i 1.00000i 0.500000 0.866025i −0.866025 + 0.500000i
835.1 1.00000i 0 0 0.500000 0.866025i 0 0.866025 0.500000i 1.00000i 0.500000 + 0.866025i −0.866025 0.500000i
835.2 1.00000i 0 0 0.500000 0.866025i 0 −0.866025 + 0.500000i 1.00000i 0.500000 + 0.866025i 0.866025 + 0.500000i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.b odd 2 1 inner
91.h even 3 1 inner
1001.r odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1001.1.r.a 4
7.c even 3 1 1001.1.bo.a yes 4
11.b odd 2 1 inner 1001.1.r.a 4
13.c even 3 1 1001.1.bo.a yes 4
77.h odd 6 1 1001.1.bo.a yes 4
91.h even 3 1 inner 1001.1.r.a 4
143.k odd 6 1 1001.1.bo.a yes 4
1001.r odd 6 1 inner 1001.1.r.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1001.1.r.a 4 1.a even 1 1 trivial
1001.1.r.a 4 11.b odd 2 1 inner
1001.1.r.a 4 91.h even 3 1 inner
1001.1.r.a 4 1001.r odd 6 1 inner
1001.1.bo.a yes 4 7.c even 3 1
1001.1.bo.a yes 4 13.c even 3 1
1001.1.bo.a yes 4 77.h odd 6 1
1001.1.bo.a yes 4 143.k odd 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(1001, [\chi])\).

Hecke characteristic polynomials

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