Properties

Label 1573.1.l.b
Level $1573$
Weight $1$
Character orbit 1573.l
Analytic conductor $0.785$
Analytic rank $0$
Dimension $4$
Projective image $D_{5}$
CM discriminant -143
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1573,1,Mod(233,1573)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1573, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([1, 5]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1573.233");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1573 = 11^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1573.l (of order \(10\), degree \(4\), 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: \(0.785029264872\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{10})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} + x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 143)
Projective image: \(D_{5}\)
Projective field: Galois closure of 5.1.20449.1
Artin image: $C_5\times D_5$
Artin field: Galois closure of 10.0.875489472034463.2

$q$-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.

\(f(q)\) \(=\) \( q + (\zeta_{10}^{2} - \zeta_{10}) q^{2} + (\zeta_{10}^{4} + 1) q^{3} + (\zeta_{10}^{4} - \zeta_{10}^{3} + \zeta_{10}^{2}) q^{4} + (\zeta_{10}^{2} - \zeta_{10} + 1) q^{6} + (\zeta_{10}^{4} + \zeta_{10}^{2}) q^{7} + (\zeta_{10}^{4} - \zeta_{10}^{3} - \zeta_{10} - 1) q^{8} + (\zeta_{10}^{4} - \zeta_{10}^{3} + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\zeta_{10}^{2} - \zeta_{10}) q^{2} + (\zeta_{10}^{4} + 1) q^{3} + (\zeta_{10}^{4} - \zeta_{10}^{3} + \zeta_{10}^{2}) q^{4} + (\zeta_{10}^{2} - \zeta_{10} + 1) q^{6} + (\zeta_{10}^{4} + \zeta_{10}^{2}) q^{7} + (\zeta_{10}^{4} - \zeta_{10}^{3} - \zeta_{10} - 1) q^{8} + (\zeta_{10}^{4} - \zeta_{10}^{3} + 1) q^{9} + (\zeta_{10}^{4} - \zeta_{10}^{3} + \zeta_{10}^{2} - \zeta_{10}) q^{12} + \zeta_{10}^{4} q^{13} + (\zeta_{10}^{4} - \zeta_{10}^{3} - \zeta_{10} + 1) q^{14} + (\zeta_{10}^{4} - \zeta_{10}^{3} - \zeta_{10}^{2} + \zeta_{10} - 1) q^{16} + (\zeta_{10}^{4} + \zeta_{10}^{2} - \zeta_{10} + 2) q^{18} + (\zeta_{10}^{4} + 1) q^{19} + (\zeta_{10}^{4} - \zeta_{10}^{3} + \zeta_{10}^{2} - \zeta_{10}) q^{21} + (\zeta_{10}^{4} - \zeta_{10}) q^{23} + (2 \zeta_{10}^{4} - \zeta_{10}^{3} + \zeta_{10}^{2} - \zeta_{10} + 1) q^{24} + \zeta_{10}^{2} q^{25} + ( - \zeta_{10} + 1) q^{26} + (\zeta_{10}^{4} + \zeta_{10}^{3} + \zeta_{10}^{2} + 1) q^{27} + (\zeta_{10}^{4} - \zeta_{10}^{3} + \zeta_{10}^{2} - 2 \zeta_{10} + 1) q^{28} + (\zeta_{10}^{4} - \zeta_{10}^{3} - \zeta_{10}^{2} + \zeta_{10} + 2) q^{32} + (\zeta_{10}^{4} - \zeta_{10}^{3} + 2 \zeta_{10}^{2} - 2 \zeta_{10} + 1) q^{36} + (\zeta_{10}^{2} - \zeta_{10} + 1) q^{38} + (\zeta_{10}^{4} - \zeta_{10}^{3}) q^{39} + ( - \zeta_{10}^{3} - \zeta_{10}) q^{41} + (2 \zeta_{10}^{4} - 2 \zeta_{10}^{3} + \zeta_{10}^{2} - \zeta_{10} + 2) q^{42} + ( - \zeta_{10}^{3} + \zeta_{10}^{2} - \zeta_{10} + 1) q^{46} + (2 \zeta_{10}^{4} - \zeta_{10}^{3} + \zeta_{10}^{2} - \zeta_{10} + 1) q^{48} + (\zeta_{10}^{4} - \zeta_{10}^{3} - \zeta_{10}) q^{49} + (\zeta_{10}^{4} - \zeta_{10}^{3}) q^{50} + ( - \zeta_{10}^{3} + \zeta_{10}^{2} - \zeta_{10}) q^{52} + (\zeta_{10}^{2} - \zeta_{10}) q^{53} + (2 \zeta_{10}^{4} - \zeta_{10}^{3} + \zeta_{10}^{2} - \zeta_{10} + 2) q^{54} + (\zeta_{10}^{4} - \zeta_{10}^{3} + \zeta_{10}^{2} - \zeta_{10} + 2) q^{56} + (2 \zeta_{10}^{4} - \zeta_{10}^{3} + 1) q^{57} + (\zeta_{10}^{4} - \zeta_{10}^{3} + \zeta_{10}^{2} - \zeta_{10} + 1) q^{63} + (\zeta_{10}^{4} - \zeta_{10}^{3} - \zeta_{10}^{2} + \zeta_{10} - 1) q^{64} + (\zeta_{10}^{4} - \zeta_{10}^{3} - \zeta_{10} + 1) q^{69} + (3 \zeta_{10}^{4} - \zeta_{10}^{3} + 2 \zeta_{10}^{2} - \zeta_{10} + 1) q^{72} + ( - \zeta_{10} + 1) q^{73} + (\zeta_{10}^{2} - \zeta_{10}) q^{75} + (\zeta_{10}^{4} - \zeta_{10}^{3} + \zeta_{10}^{2} - \zeta_{10}) q^{76} + (\zeta_{10}^{4} - \zeta_{10} + 2) q^{78} + (\zeta_{10}^{4} + \zeta_{10}^{3} - \zeta_{10}^{2} - \zeta_{10} + 1) q^{81} + (\zeta_{10}^{4} - \zeta_{10}^{3} + \zeta_{10}^{2} + 1) q^{82} + (\zeta_{10}^{2} + 1) q^{83} + (2 \zeta_{10}^{4} - \zeta_{10}^{3} + 2 \zeta_{10}^{2} - 3 \zeta_{10} + 3) q^{84} + ( - \zeta_{10}^{3} - \zeta_{10}) q^{91} + (\zeta_{10}^{4} - \zeta_{10}^{3} + \zeta_{10}^{2} - \zeta_{10} + 1) q^{92} + (3 \zeta_{10}^{4} - \zeta_{10}^{3} + \zeta_{10}^{2} - \zeta_{10} + 1) q^{96} + (\zeta_{10}^{4} - \zeta_{10}^{3} + \zeta_{10}^{2} - \zeta_{10} + 2) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{2} + 3 q^{3} - 3 q^{4} + q^{6} - 2 q^{7} + q^{8} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 2 q^{2} + 3 q^{3} - 3 q^{4} + q^{6} - 2 q^{7} + q^{8} + 2 q^{9} - 6 q^{12} - q^{13} + q^{14} + 4 q^{18} + 3 q^{19} - 4 q^{21} - 2 q^{23} + 2 q^{24} - q^{25} + 3 q^{26} + q^{27} - q^{28} + 4 q^{32} - 4 q^{36} + q^{38} - 2 q^{39} - 2 q^{41} + 2 q^{42} + q^{46} - 3 q^{49} - 2 q^{50} - 3 q^{52} - 2 q^{53} + 2 q^{54} + 2 q^{56} + q^{57} - q^{63} - 2 q^{64} + q^{69} - 2 q^{72} + 3 q^{73} - 2 q^{75} - 6 q^{76} + 6 q^{78} + q^{82} + 3 q^{83} + 3 q^{84} - 2 q^{91} - q^{92} + 3 q^{96} + 4 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1573\mathbb{Z}\right)^\times\).

\(n\) \(365\) \(1211\)
\(\chi(n)\) \(\zeta_{10}^{3}\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
233.1
−0.309017 0.951057i
0.809017 0.587785i
0.809017 + 0.587785i
−0.309017 + 0.951057i
−0.500000 + 1.53884i 1.30902 0.951057i −1.30902 0.951057i 0 0.809017 + 2.48990i −0.500000 0.363271i 0.809017 0.587785i 0.500000 1.53884i 0
766.1 −0.500000 0.363271i 0.190983 0.587785i −0.190983 0.587785i 0 −0.309017 + 0.224514i −0.500000 1.53884i −0.309017 + 0.951057i 0.500000 + 0.363271i 0
844.1 −0.500000 + 0.363271i 0.190983 + 0.587785i −0.190983 + 0.587785i 0 −0.309017 0.224514i −0.500000 + 1.53884i −0.309017 0.951057i 0.500000 0.363271i 0
1546.1 −0.500000 1.53884i 1.30902 + 0.951057i −1.30902 + 0.951057i 0 0.809017 2.48990i −0.500000 + 0.363271i 0.809017 + 0.587785i 0.500000 + 1.53884i 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
143.d odd 2 1 CM by \(\Q(\sqrt{-143}) \)
11.c even 5 1 inner
143.l odd 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1573.1.l.b 4
11.b odd 2 1 1573.1.l.c 4
11.c even 5 1 143.1.d.a 2
11.c even 5 1 inner 1573.1.l.b 4
11.c even 5 2 1573.1.l.d 4
11.d odd 10 1 143.1.d.b yes 2
11.d odd 10 2 1573.1.l.a 4
11.d odd 10 1 1573.1.l.c 4
13.b even 2 1 1573.1.l.c 4
33.f even 10 1 1287.1.g.a 2
33.h odd 10 1 1287.1.g.b 2
44.g even 10 1 2288.1.m.a 2
44.h odd 10 1 2288.1.m.b 2
55.h odd 10 1 3575.1.h.e 2
55.j even 10 1 3575.1.h.f 2
55.k odd 20 2 3575.1.c.d 4
55.l even 20 2 3575.1.c.c 4
143.d odd 2 1 CM 1573.1.l.b 4
143.l odd 10 1 143.1.d.a 2
143.l odd 10 1 inner 1573.1.l.b 4
143.l odd 10 2 1573.1.l.d 4
143.n even 10 1 143.1.d.b yes 2
143.n even 10 2 1573.1.l.a 4
143.n even 10 1 1573.1.l.c 4
143.q even 15 2 1859.1.i.b 4
143.r odd 20 2 1859.1.c.c 4
143.s even 20 2 1859.1.c.c 4
143.t odd 30 2 1859.1.i.a 4
143.u even 30 2 1859.1.i.a 4
143.v odd 30 2 1859.1.i.b 4
143.w even 60 4 1859.1.k.c 8
143.x odd 60 4 1859.1.k.c 8
429.v odd 10 1 1287.1.g.a 2
429.y even 10 1 1287.1.g.b 2
572.ba odd 10 1 2288.1.m.a 2
572.bb even 10 1 2288.1.m.b 2
715.bf even 10 1 3575.1.h.e 2
715.bi odd 10 1 3575.1.h.f 2
715.cb even 20 2 3575.1.c.d 4
715.ce odd 20 2 3575.1.c.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
143.1.d.a 2 11.c even 5 1
143.1.d.a 2 143.l odd 10 1
143.1.d.b yes 2 11.d odd 10 1
143.1.d.b yes 2 143.n even 10 1
1287.1.g.a 2 33.f even 10 1
1287.1.g.a 2 429.v odd 10 1
1287.1.g.b 2 33.h odd 10 1
1287.1.g.b 2 429.y even 10 1
1573.1.l.a 4 11.d odd 10 2
1573.1.l.a 4 143.n even 10 2
1573.1.l.b 4 1.a even 1 1 trivial
1573.1.l.b 4 11.c even 5 1 inner
1573.1.l.b 4 143.d odd 2 1 CM
1573.1.l.b 4 143.l odd 10 1 inner
1573.1.l.c 4 11.b odd 2 1
1573.1.l.c 4 11.d odd 10 1
1573.1.l.c 4 13.b even 2 1
1573.1.l.c 4 143.n even 10 1
1573.1.l.d 4 11.c even 5 2
1573.1.l.d 4 143.l odd 10 2
1859.1.c.c 4 143.r odd 20 2
1859.1.c.c 4 143.s even 20 2
1859.1.i.a 4 143.t odd 30 2
1859.1.i.a 4 143.u even 30 2
1859.1.i.b 4 143.q even 15 2
1859.1.i.b 4 143.v odd 30 2
1859.1.k.c 8 143.w even 60 4
1859.1.k.c 8 143.x odd 60 4
2288.1.m.a 2 44.g even 10 1
2288.1.m.a 2 572.ba odd 10 1
2288.1.m.b 2 44.h odd 10 1
2288.1.m.b 2 572.bb even 10 1
3575.1.c.c 4 55.l even 20 2
3575.1.c.c 4 715.ce odd 20 2
3575.1.c.d 4 55.k odd 20 2
3575.1.c.d 4 715.cb even 20 2
3575.1.h.e 2 55.h odd 10 1
3575.1.h.e 2 715.bf even 10 1
3575.1.h.f 2 55.j even 10 1
3575.1.h.f 2 715.bi odd 10 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} + 2T_{2}^{3} + 4T_{2}^{2} + 3T_{2} + 1 \) acting on \(S_{1}^{\mathrm{new}}(1573, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

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