Properties

Label 532.1.n.a
Level $532$
Weight $1$
Character orbit 532.n
Analytic conductor $0.266$
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] = [532,1,Mod(163,532)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(532, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 2, 4]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("532.163");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 532 = 2^{2} \cdot 7 \cdot 19 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 532.n (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: \(0.265502586721\)
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: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(A_{4}\)
Projective field: Galois closure of 4.0.283024.1

$q$-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.

\(f(q)\) \(=\) \( q + \zeta_{12}^{5} q^{2} - \zeta_{12}^{3} q^{3} - \zeta_{12}^{4} q^{4} + \zeta_{12}^{2} q^{6} + \zeta_{12}^{3} q^{7} + \zeta_{12}^{3} q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{12}^{5} q^{2} - \zeta_{12}^{3} q^{3} - \zeta_{12}^{4} q^{4} + \zeta_{12}^{2} q^{6} + \zeta_{12}^{3} q^{7} + \zeta_{12}^{3} q^{8} - \zeta_{12}^{5} q^{11} - \zeta_{12} q^{12} + \zeta_{12}^{2} q^{13} - \zeta_{12}^{2} q^{14} - \zeta_{12}^{2} q^{16} - q^{17} - \zeta_{12}^{5} q^{19} + q^{21} + \zeta_{12}^{4} q^{22} - \zeta_{12}^{3} q^{23} + q^{24} - \zeta_{12}^{4} q^{25} - \zeta_{12} q^{26} - \zeta_{12}^{3} q^{27} + \zeta_{12} q^{28} - \zeta_{12}^{2} q^{29} + \zeta_{12}^{5} q^{31} + \zeta_{12} q^{32} - \zeta_{12}^{2} q^{33} - \zeta_{12}^{5} q^{34} + \zeta_{12}^{4} q^{37} + \zeta_{12}^{4} q^{38} - \zeta_{12}^{5} q^{39} + \zeta_{12}^{4} q^{41} + \zeta_{12}^{5} q^{42} + \zeta_{12} q^{43} - \zeta_{12}^{3} q^{44} + \zeta_{12}^{2} q^{46} + \zeta_{12}^{3} q^{47} + \zeta_{12}^{5} q^{48} - q^{49} + \zeta_{12}^{3} q^{50} + \zeta_{12}^{3} q^{51} + q^{52} + \zeta_{12}^{2} q^{54} - q^{56} - \zeta_{12}^{2} q^{57} + \zeta_{12} q^{58} + \zeta_{12}^{3} q^{59} + q^{61} - \zeta_{12}^{4} q^{62} - q^{64} + \zeta_{12} q^{66} + \zeta_{12}^{4} q^{68} - q^{69} - \zeta_{12} q^{71} - q^{73} - \zeta_{12}^{3} q^{74} - \zeta_{12} q^{75} - \zeta_{12}^{3} q^{76} + \zeta_{12}^{2} q^{77} + \zeta_{12}^{4} q^{78} - q^{81} - \zeta_{12}^{3} q^{82} - \zeta_{12}^{4} q^{84} - q^{86} + \zeta_{12}^{5} q^{87} + \zeta_{12}^{2} q^{88} - q^{89} + \zeta_{12}^{5} q^{91} - \zeta_{12} q^{92} + \zeta_{12}^{2} q^{93} - \zeta_{12}^{2} q^{94} - \zeta_{12}^{4} q^{96} - \zeta_{12}^{4} q^{97} - \zeta_{12}^{5} q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{4} + 2 q^{6}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{4} + 2 q^{6} + 2 q^{13} - 2 q^{14} - 2 q^{16} - 4 q^{17} + 4 q^{21} - 2 q^{22} + 4 q^{24} + 2 q^{25} - 2 q^{29} - 2 q^{33} - 2 q^{37} - 2 q^{38} - 2 q^{41} + 2 q^{46} - 4 q^{49} + 4 q^{52} + 2 q^{54} - 4 q^{56} - 2 q^{57} + 4 q^{61} + 2 q^{62} - 4 q^{64} - 2 q^{68} - 4 q^{69} - 4 q^{73} + 2 q^{77} - 2 q^{78} - 4 q^{81} + 2 q^{84} - 4 q^{86} + 2 q^{88} - 4 q^{89} + 2 q^{93} - 2 q^{94} + 2 q^{96} + 2 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/532\mathbb{Z}\right)^\times\).

\(n\) \(267\) \(381\) \(477\)
\(\chi(n)\) \(-1\) \(\zeta_{12}^{4}\) \(-\zeta_{12}^{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
163.1
0.866025 + 0.500000i
−0.866025 0.500000i
0.866025 0.500000i
−0.866025 + 0.500000i
−0.866025 + 0.500000i 1.00000i 0.500000 0.866025i 0 0.500000 + 0.866025i 1.00000i 1.00000i 0 0
163.2 0.866025 0.500000i 1.00000i 0.500000 0.866025i 0 0.500000 + 0.866025i 1.00000i 1.00000i 0 0
235.1 −0.866025 0.500000i 1.00000i 0.500000 + 0.866025i 0 0.500000 0.866025i 1.00000i 1.00000i 0 0
235.2 0.866025 + 0.500000i 1.00000i 0.500000 + 0.866025i 0 0.500000 0.866025i 1.00000i 1.00000i 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
133.g even 3 1 inner
532.n odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 532.1.n.a 4
4.b odd 2 1 inner 532.1.n.a 4
7.b odd 2 1 3724.1.n.a 4
7.c even 3 1 532.1.bk.a yes 4
7.c even 3 1 3724.1.bb.b 4
7.d odd 6 1 3724.1.bb.a 4
7.d odd 6 1 3724.1.bk.a 4
19.c even 3 1 532.1.bk.a yes 4
28.d even 2 1 3724.1.n.a 4
28.f even 6 1 3724.1.bb.a 4
28.f even 6 1 3724.1.bk.a 4
28.g odd 6 1 532.1.bk.a yes 4
28.g odd 6 1 3724.1.bb.b 4
76.g odd 6 1 532.1.bk.a yes 4
133.g even 3 1 inner 532.1.n.a 4
133.h even 3 1 3724.1.bb.b 4
133.k odd 6 1 3724.1.n.a 4
133.m odd 6 1 3724.1.bk.a 4
133.t odd 6 1 3724.1.bb.a 4
532.n odd 6 1 inner 532.1.n.a 4
532.r even 6 1 3724.1.bb.a 4
532.u even 6 1 3724.1.bk.a 4
532.bk odd 6 1 3724.1.bb.b 4
532.bn even 6 1 3724.1.n.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
532.1.n.a 4 1.a even 1 1 trivial
532.1.n.a 4 4.b odd 2 1 inner
532.1.n.a 4 133.g even 3 1 inner
532.1.n.a 4 532.n odd 6 1 inner
532.1.bk.a yes 4 7.c even 3 1
532.1.bk.a yes 4 19.c even 3 1
532.1.bk.a yes 4 28.g odd 6 1
532.1.bk.a yes 4 76.g odd 6 1
3724.1.n.a 4 7.b odd 2 1
3724.1.n.a 4 28.d even 2 1
3724.1.n.a 4 133.k odd 6 1
3724.1.n.a 4 532.bn even 6 1
3724.1.bb.a 4 7.d odd 6 1
3724.1.bb.a 4 28.f even 6 1
3724.1.bb.a 4 133.t odd 6 1
3724.1.bb.a 4 532.r even 6 1
3724.1.bb.b 4 7.c even 3 1
3724.1.bb.b 4 28.g odd 6 1
3724.1.bb.b 4 133.h even 3 1
3724.1.bb.b 4 532.bk odd 6 1
3724.1.bk.a 4 7.d odd 6 1
3724.1.bk.a 4 28.f even 6 1
3724.1.bk.a 4 133.m odd 6 1
3724.1.bk.a 4 532.u even 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(532, [\chi])\).

Hecke characteristic polynomials

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