Properties

Label 2523.1.h.b
Level $2523$
Weight $1$
Character orbit 2523.h
Analytic conductor $1.259$
Analytic rank $0$
Dimension $6$
Projective image $D_{3}$
CM discriminant -87
Inner twists $12$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2523,1,Mod(236,2523)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2523, base_ring=CyclotomicField(14))
 
chi = DirichletCharacter(H, H._module([7, 1]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2523.236");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2523 = 3 \cdot 29^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2523.h (of order \(14\), degree \(6\), 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: \(1.25914102687\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\Q(\zeta_{14})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{5} + x^{4} - x^{3} + x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 87)
Projective image: \(D_{3}\)
Projective field: Galois closure of 3.1.87.1
Artin image: $S_3\times C_7$
Artin field: Galois closure of 21.7.13347832346292311387708944226103.1

$q$-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.

\(f(q)\) \(=\) \( q - \zeta_{14}^{6} q^{2} + \zeta_{14}^{2} q^{3} + \zeta_{14} q^{6} - \zeta_{14}^{2} q^{7} + \zeta_{14}^{4} q^{8} + \zeta_{14}^{4} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - \zeta_{14}^{6} q^{2} + \zeta_{14}^{2} q^{3} + \zeta_{14} q^{6} - \zeta_{14}^{2} q^{7} + \zeta_{14}^{4} q^{8} + \zeta_{14}^{4} q^{9} + \zeta_{14}^{3} q^{11} + \zeta_{14}^{3} q^{13} - \zeta_{14} q^{14} + \zeta_{14}^{3} q^{16} - q^{17} + \zeta_{14}^{3} q^{18} - \zeta_{14}^{4} q^{21} + \zeta_{14}^{2} q^{22} + \zeta_{14}^{6} q^{24} - \zeta_{14}^{5} q^{25} + \zeta_{14}^{2} q^{26} + \zeta_{14}^{6} q^{27} + \zeta_{14}^{5} q^{33} + \zeta_{14}^{6} q^{34} + \zeta_{14}^{5} q^{39} + 2 q^{41} - \zeta_{14}^{3} q^{42} + \zeta_{14}^{3} q^{47} + \zeta_{14}^{5} q^{48} - \zeta_{14}^{4} q^{50} - \zeta_{14}^{2} q^{51} + \zeta_{14}^{5} q^{54} - \zeta_{14}^{6} q^{56} - \zeta_{14}^{6} q^{63} - \zeta_{14} q^{64} + \zeta_{14}^{4} q^{66} - \zeta_{14}^{4} q^{67} - \zeta_{14} q^{72} + q^{75} - \zeta_{14}^{5} q^{77} + \zeta_{14}^{4} q^{78} - \zeta_{14} q^{81} - 2 \zeta_{14}^{6} q^{82} - q^{88} - \zeta_{14}^{6} q^{89} - \zeta_{14}^{5} q^{91} + \zeta_{14}^{2} q^{94} - q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + q^{2} - q^{3} + q^{6} + q^{7} - q^{8} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + q^{2} - q^{3} + q^{6} + q^{7} - q^{8} - q^{9} + q^{11} + q^{13} - q^{14} + q^{16} - 6 q^{17} + q^{18} + q^{21} - q^{22} - q^{24} - q^{25} - q^{26} - q^{27} + q^{33} - q^{34} + q^{39} + 12 q^{41} - q^{42} + q^{47} + q^{48} + q^{50} + q^{51} + q^{54} + q^{56} + q^{63} - q^{64} - q^{66} + q^{67} - q^{72} + 6 q^{75} - q^{77} - q^{78} - q^{81} + 2 q^{82} - 6 q^{88} + q^{89} - q^{91} - q^{94} - 6 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2523\mathbb{Z}\right)^\times\).

\(n\) \(842\) \(1684\)
\(\chi(n)\) \(-1\) \(\zeta_{14}^{5}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
236.1
0.222521 + 0.974928i
0.222521 0.974928i
−0.623490 + 0.781831i
−0.623490 0.781831i
0.900969 + 0.433884i
0.900969 0.433884i
0.222521 0.974928i −0.900969 + 0.433884i 0 0 0.222521 + 0.974928i 0.900969 0.433884i 0.623490 0.781831i 0.623490 0.781831i 0
1037.1 0.222521 + 0.974928i −0.900969 0.433884i 0 0 0.222521 0.974928i 0.900969 + 0.433884i 0.623490 + 0.781831i 0.623490 + 0.781831i 0
1745.1 −0.623490 0.781831i −0.222521 0.974928i 0 0 −0.623490 + 0.781831i 0.222521 + 0.974928i −0.900969 + 0.433884i −0.900969 + 0.433884i 0
1949.1 −0.623490 + 0.781831i −0.222521 + 0.974928i 0 0 −0.623490 0.781831i 0.222521 0.974928i −0.900969 0.433884i −0.900969 0.433884i 0
1952.1 0.900969 0.433884i 0.623490 + 0.781831i 0 0 0.900969 + 0.433884i −0.623490 0.781831i −0.222521 + 0.974928i −0.222521 + 0.974928i 0
2333.1 0.900969 + 0.433884i 0.623490 0.781831i 0 0 0.900969 0.433884i −0.623490 + 0.781831i −0.222521 0.974928i −0.222521 0.974928i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 236.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
87.d odd 2 1 CM by \(\Q(\sqrt{-87}) \)
29.d even 7 5 inner
87.h odd 14 5 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2523.1.h.b 6
3.b odd 2 1 2523.1.h.a 6
29.b even 2 1 2523.1.h.a 6
29.c odd 4 2 2523.1.j.b 12
29.d even 7 1 87.1.d.a 1
29.d even 7 5 inner 2523.1.h.b 6
29.e even 14 1 87.1.d.b yes 1
29.e even 14 5 2523.1.h.a 6
29.f odd 28 2 2523.1.b.b 2
29.f odd 28 10 2523.1.j.b 12
87.d odd 2 1 CM 2523.1.h.b 6
87.f even 4 2 2523.1.j.b 12
87.h odd 14 1 87.1.d.a 1
87.h odd 14 5 inner 2523.1.h.b 6
87.j odd 14 1 87.1.d.b yes 1
87.j odd 14 5 2523.1.h.a 6
87.k even 28 2 2523.1.b.b 2
87.k even 28 10 2523.1.j.b 12
116.h odd 14 1 1392.1.i.b 1
116.j odd 14 1 1392.1.i.a 1
145.l even 14 1 2175.1.h.a 1
145.n even 14 1 2175.1.h.b 1
145.p odd 28 2 2175.1.b.a 2
145.q odd 28 2 2175.1.b.b 2
261.q even 21 2 2349.1.h.b 2
261.t odd 42 2 2349.1.h.a 2
261.u even 42 2 2349.1.h.a 2
261.v odd 42 2 2349.1.h.b 2
348.s even 14 1 1392.1.i.b 1
348.t even 14 1 1392.1.i.a 1
435.w odd 14 1 2175.1.h.a 1
435.bb odd 14 1 2175.1.h.b 1
435.bg even 28 2 2175.1.b.a 2
435.bj even 28 2 2175.1.b.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
87.1.d.a 1 29.d even 7 1
87.1.d.a 1 87.h odd 14 1
87.1.d.b yes 1 29.e even 14 1
87.1.d.b yes 1 87.j odd 14 1
1392.1.i.a 1 116.j odd 14 1
1392.1.i.a 1 348.t even 14 1
1392.1.i.b 1 116.h odd 14 1
1392.1.i.b 1 348.s even 14 1
2175.1.b.a 2 145.p odd 28 2
2175.1.b.a 2 435.bg even 28 2
2175.1.b.b 2 145.q odd 28 2
2175.1.b.b 2 435.bj even 28 2
2175.1.h.a 1 145.l even 14 1
2175.1.h.a 1 435.w odd 14 1
2175.1.h.b 1 145.n even 14 1
2175.1.h.b 1 435.bb odd 14 1
2349.1.h.a 2 261.t odd 42 2
2349.1.h.a 2 261.u even 42 2
2349.1.h.b 2 261.q even 21 2
2349.1.h.b 2 261.v odd 42 2
2523.1.b.b 2 29.f odd 28 2
2523.1.b.b 2 87.k even 28 2
2523.1.h.a 6 3.b odd 2 1
2523.1.h.a 6 29.b even 2 1
2523.1.h.a 6 29.e even 14 5
2523.1.h.a 6 87.j odd 14 5
2523.1.h.b 6 1.a even 1 1 trivial
2523.1.h.b 6 29.d even 7 5 inner
2523.1.h.b 6 87.d odd 2 1 CM
2523.1.h.b 6 87.h odd 14 5 inner
2523.1.j.b 12 29.c odd 4 2
2523.1.j.b 12 29.f odd 28 10
2523.1.j.b 12 87.f even 4 2
2523.1.j.b 12 87.k even 28 10

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{6} - T_{2}^{5} + T_{2}^{4} - T_{2}^{3} + T_{2}^{2} - T_{2} + 1 \) acting on \(S_{1}^{\mathrm{new}}(2523, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

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