Properties

Label 2349.1.h.b
Level $2349$
Weight $1$
Character orbit 2349.h
Analytic conductor $1.172$
Analytic rank $0$
Dimension $2$
Projective image $D_{3}$
CM discriminant -87
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2349,1,Mod(782,2349)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2349, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([5, 3]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2349.782");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2349 = 3^{4} \cdot 29 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2349.h (of order \(6\), degree \(2\), 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: \(1.17230371467\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) 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: $C_3\times S_3$
Artin field: Galois closure of 6.0.480048687.2

$q$-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.

\(f(q)\) \(=\) \( q + \zeta_{6} q^{2} + \zeta_{6} q^{7} + q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{6} q^{2} + \zeta_{6} q^{7} + q^{8} + \zeta_{6} q^{11} - \zeta_{6}^{2} q^{13} + \zeta_{6}^{2} q^{14} + \zeta_{6} q^{16} - q^{17} + \zeta_{6}^{2} q^{22} - \zeta_{6} q^{25} + q^{26} - \zeta_{6} q^{29} - \zeta_{6} q^{34} + 2 \zeta_{6}^{2} q^{41} + \zeta_{6} q^{47} - \zeta_{6}^{2} q^{50} + \zeta_{6} q^{56} - \zeta_{6}^{2} q^{58} + q^{64} - \zeta_{6}^{2} q^{67} + \zeta_{6}^{2} q^{77} - 2 q^{82} + \zeta_{6} q^{88} - q^{89} + q^{91} + \zeta_{6}^{2} q^{94} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{2} + q^{7} + 2 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + q^{2} + q^{7} + 2 q^{8} + q^{11} + q^{13} - q^{14} + q^{16} - 2 q^{17} - q^{22} - q^{25} + 2 q^{26} - q^{29} - q^{34} - 2 q^{41} + q^{47} + q^{50} + q^{56} + q^{58} + 2 q^{64} + q^{67} - q^{77} - 4 q^{82} + q^{88} - 2 q^{89} + 2 q^{91} - q^{94}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2349\mathbb{Z}\right)^\times\).

\(n\) \(407\) \(1945\)
\(\chi(n)\) \(-\zeta_{6}^{2}\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
782.1
0.500000 + 0.866025i
0.500000 0.866025i
0.500000 + 0.866025i 0 0 0 0 0.500000 + 0.866025i 1.00000 0 0
1565.1 0.500000 0.866025i 0 0 0 0 0.500000 0.866025i 1.00000 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
87.d odd 2 1 CM by \(\Q(\sqrt{-87}) \)
9.c even 3 1 inner
261.h odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2349.1.h.b 2
3.b odd 2 1 2349.1.h.a 2
9.c even 3 1 87.1.d.a 1
9.c even 3 1 inner 2349.1.h.b 2
9.d odd 6 1 87.1.d.b yes 1
9.d odd 6 1 2349.1.h.a 2
29.b even 2 1 2349.1.h.a 2
36.f odd 6 1 1392.1.i.a 1
36.h even 6 1 1392.1.i.b 1
45.h odd 6 1 2175.1.h.a 1
45.j even 6 1 2175.1.h.b 1
45.k odd 12 2 2175.1.b.a 2
45.l even 12 2 2175.1.b.b 2
87.d odd 2 1 CM 2349.1.h.b 2
261.h odd 6 1 87.1.d.a 1
261.h odd 6 1 inner 2349.1.h.b 2
261.i even 6 1 87.1.d.b yes 1
261.i even 6 1 2349.1.h.a 2
261.l even 12 2 2523.1.b.b 2
261.m odd 12 2 2523.1.b.b 2
261.q even 21 6 2523.1.h.b 6
261.t odd 42 6 2523.1.h.a 6
261.u even 42 6 2523.1.h.a 6
261.v odd 42 6 2523.1.h.b 6
261.w odd 84 12 2523.1.j.b 12
261.x even 84 12 2523.1.j.b 12
1044.o odd 6 1 1392.1.i.b 1
1044.t even 6 1 1392.1.i.a 1
1305.w even 6 1 2175.1.h.a 1
1305.ba odd 6 1 2175.1.h.b 1
1305.bi even 12 2 2175.1.b.a 2
1305.bk odd 12 2 2175.1.b.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
87.1.d.a 1 9.c even 3 1
87.1.d.a 1 261.h odd 6 1
87.1.d.b yes 1 9.d odd 6 1
87.1.d.b yes 1 261.i even 6 1
1392.1.i.a 1 36.f odd 6 1
1392.1.i.a 1 1044.t even 6 1
1392.1.i.b 1 36.h even 6 1
1392.1.i.b 1 1044.o odd 6 1
2175.1.b.a 2 45.k odd 12 2
2175.1.b.a 2 1305.bi even 12 2
2175.1.b.b 2 45.l even 12 2
2175.1.b.b 2 1305.bk odd 12 2
2175.1.h.a 1 45.h odd 6 1
2175.1.h.a 1 1305.w even 6 1
2175.1.h.b 1 45.j even 6 1
2175.1.h.b 1 1305.ba odd 6 1
2349.1.h.a 2 3.b odd 2 1
2349.1.h.a 2 9.d odd 6 1
2349.1.h.a 2 29.b even 2 1
2349.1.h.a 2 261.i even 6 1
2349.1.h.b 2 1.a even 1 1 trivial
2349.1.h.b 2 9.c even 3 1 inner
2349.1.h.b 2 87.d odd 2 1 CM
2349.1.h.b 2 261.h odd 6 1 inner
2523.1.b.b 2 261.l even 12 2
2523.1.b.b 2 261.m odd 12 2
2523.1.h.a 6 261.t odd 42 6
2523.1.h.a 6 261.u even 42 6
2523.1.h.b 6 261.q even 21 6
2523.1.h.b 6 261.v odd 42 6
2523.1.j.b 12 261.w odd 84 12
2523.1.j.b 12 261.x even 84 12

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} - T_{2} + 1 \) acting on \(S_{1}^{\mathrm{new}}(2349, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

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