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Properties

Label 3549.1.cv.b
Level $3549$
Weight $1$
Character orbit 3549.cv
Analytic conductor $1.771$
Analytic rank $0$
Dimension $24$
Projective image $D_{52}$
CM discriminant -3
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3549,1,Mod(83,3549)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3549, base_ring=CyclotomicField(52))
 
chi = DirichletCharacter(H, H._module([26, 26, 15]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3549.83");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3549 = 3 \cdot 7 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3549.cv (of order \(52\), degree \(24\), 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: \(1.77118172983\)
Analytic rank: \(0\)
Dimension: \(24\)
Coefficient field: \(\Q(\zeta_{52})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{24} - x^{22} + x^{20} - x^{18} + x^{16} - x^{14} + x^{12} - x^{10} + x^{8} - x^{6} + x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{52}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{52} - \cdots)\)

$q$-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.

\(f(q)\) \(=\) \( q + \zeta_{52}^{11} q^{3} - \zeta_{52}^{5} q^{4} - \zeta_{52}^{3} q^{7} + \zeta_{52}^{22} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{52}^{11} q^{3} - \zeta_{52}^{5} q^{4} - \zeta_{52}^{3} q^{7} + \zeta_{52}^{22} q^{9} - \zeta_{52}^{16} q^{12} - \zeta_{52}^{13} q^{13} + \zeta_{52}^{10} q^{16} + (\zeta_{52}^{9} + \zeta_{52}^{4}) q^{19} - \zeta_{52}^{14} q^{21} - \zeta_{52}^{19} q^{25} - \zeta_{52}^{7} q^{27} + \zeta_{52}^{8} q^{28} + (\zeta_{52}^{25} - \zeta_{52}^{2}) q^{31} + \zeta_{52} q^{36} + ( - \zeta_{52}^{21} + \zeta_{52}^{6}) q^{37} - \zeta_{52}^{24} q^{39} + ( - \zeta_{52}^{20} - \zeta_{52}^{18}) q^{43} + \zeta_{52}^{21} q^{48} + \zeta_{52}^{6} q^{49} + \zeta_{52}^{18} q^{52} + (\zeta_{52}^{20} + \zeta_{52}^{15}) q^{57} + ( - \zeta_{52}^{17} + \zeta_{52}^{7}) q^{61} - \zeta_{52}^{25} q^{63} - \zeta_{52}^{15} q^{64} + (\zeta_{52}^{2} + \zeta_{52}) q^{67} + ( - \zeta_{52}^{20} + \zeta_{52}) q^{73} + \zeta_{52}^{4} q^{75} + ( - \zeta_{52}^{14} - \zeta_{52}^{9}) q^{76} + ( - \zeta_{52}^{23} + \zeta_{52}^{19}) q^{79} - \zeta_{52}^{18} q^{81} + \zeta_{52}^{19} q^{84} + \zeta_{52}^{16} q^{91} + ( - \zeta_{52}^{13} - \zeta_{52}^{10}) q^{93} + (\zeta_{52}^{23} + \zeta_{52}^{10}) q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 24 q + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 24 q + 2 q^{9} + 2 q^{12} + 2 q^{16} - 2 q^{19} - 2 q^{21} - 2 q^{28} - 2 q^{31} + 2 q^{37} + 2 q^{39} + 2 q^{49} + 2 q^{52} - 2 q^{57} + 2 q^{67} + 2 q^{73} - 2 q^{75} - 2 q^{76} - 2 q^{81} - 2 q^{91} - 2 q^{93} + 2 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3549\mathbb{Z}\right)^\times\).

\(n\) \(1184\) \(1522\) \(3382\)
\(\chi(n)\) \(-1\) \(-1\) \(-\zeta_{52}^{5}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
83.1
−0.935016 − 0.354605i
0.663123 + 0.748511i
0.992709 + 0.120537i
−0.464723 − 0.885456i
−0.992709 + 0.120537i
0.239316 + 0.970942i
0.935016 − 0.354605i
−0.822984 + 0.568065i
−0.239316 + 0.970942i
0.663123 − 0.748511i
0.464723 − 0.885456i
−0.464723 + 0.885456i
−0.663123 + 0.748511i
0.239316 − 0.970942i
0.822984 − 0.568065i
−0.935016 + 0.354605i
−0.239316 − 0.970942i
0.992709 − 0.120537i
0.464723 + 0.885456i
−0.992709 − 0.120537i
0 0.663123 + 0.748511i −0.239316 + 0.970942i 0 0 0.464723 + 0.885456i 0 −0.120537 + 0.992709i 0
125.1 0 −0.992709 + 0.120537i 0.464723 + 0.885456i 0 0 0.822984 − 0.568065i 0 0.970942 − 0.239316i 0
356.1 0 0.239316 + 0.970942i −0.822984 − 0.568065i 0 0 −0.935016 − 0.354605i 0 −0.885456 + 0.464723i 0
398.1 0 −0.822984 + 0.568065i 0.663123 − 0.748511i 0 0 −0.992709 − 0.120537i 0 0.354605 − 0.935016i 0
629.1 0 −0.239316 + 0.970942i 0.822984 − 0.568065i 0 0 0.935016 − 0.354605i 0 −0.885456 − 0.464723i 0
671.1 0 −0.464723 + 0.885456i −0.935016 − 0.354605i 0 0 0.663123 + 0.748511i 0 −0.568065 − 0.822984i 0
902.1 0 −0.663123 + 0.748511i 0.239316 + 0.970942i 0 0 −0.464723 + 0.885456i 0 −0.120537 − 0.992709i 0
1175.1 0 −0.935016 + 0.354605i −0.992709 − 0.120537i 0 0 −0.239316 − 0.970942i 0 0.748511 − 0.663123i 0
1217.1 0 0.464723 + 0.885456i 0.935016 − 0.354605i 0 0 −0.663123 + 0.748511i 0 −0.568065 + 0.822984i 0
1448.1 0 −0.992709 − 0.120537i 0.464723 − 0.885456i 0 0 0.822984 + 0.568065i 0 0.970942 + 0.239316i 0
1490.1 0 0.822984 + 0.568065i −0.663123 − 0.748511i 0 0 0.992709 − 0.120537i 0 0.354605 + 0.935016i 0
1721.1 0 −0.822984 − 0.568065i 0.663123 + 0.748511i 0 0 −0.992709 + 0.120537i 0 0.354605 + 0.935016i 0
1763.1 0 0.992709 + 0.120537i −0.464723 + 0.885456i 0 0 −0.822984 − 0.568065i 0 0.970942 + 0.239316i 0
1994.1 0 −0.464723 − 0.885456i −0.935016 + 0.354605i 0 0 0.663123 − 0.748511i 0 −0.568065 + 0.822984i 0
2036.1 0 0.935016 − 0.354605i 0.992709 + 0.120537i 0 0 0.239316 + 0.970942i 0 0.748511 − 0.663123i 0
2309.1 0 0.663123 − 0.748511i −0.239316 − 0.970942i 0 0 0.464723 − 0.885456i 0 −0.120537 − 0.992709i 0
2540.1 0 0.464723 − 0.885456i 0.935016 + 0.354605i 0 0 −0.663123 − 0.748511i 0 −0.568065 − 0.822984i 0
2582.1 0 0.239316 − 0.970942i −0.822984 + 0.568065i 0 0 −0.935016 + 0.354605i 0 −0.885456 − 0.464723i 0
2813.1 0 0.822984 − 0.568065i −0.663123 + 0.748511i 0 0 0.992709 + 0.120537i 0 0.354605 − 0.935016i 0
2855.1 0 −0.239316 − 0.970942i 0.822984 + 0.568065i 0 0 0.935016 + 0.354605i 0 −0.885456 + 0.464723i 0
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 83.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)
1183.bn even 52 1 inner
3549.cv odd 52 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3549.1.cv.b yes 24
3.b odd 2 1 CM 3549.1.cv.b yes 24
7.b odd 2 1 3549.1.cv.a ✓ 24
21.c even 2 1 3549.1.cv.a ✓ 24
169.j odd 52 1 3549.1.cv.a ✓ 24
507.s even 52 1 3549.1.cv.a ✓ 24
1183.bn even 52 1 inner 3549.1.cv.b yes 24
3549.cv odd 52 1 inner 3549.1.cv.b yes 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3549.1.cv.a ✓ 24 7.b odd 2 1
3549.1.cv.a ✓ 24 21.c even 2 1
3549.1.cv.a ✓ 24 169.j odd 52 1
3549.1.cv.a ✓ 24 507.s even 52 1
3549.1.cv.b yes 24 1.a even 1 1 trivial
3549.1.cv.b yes 24 3.b odd 2 1 CM
3549.1.cv.b yes 24 1183.bn even 52 1 inner
3549.1.cv.b yes 24 3549.cv odd 52 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{19}^{24} + 2 T_{19}^{23} + 2 T_{19}^{22} + 35 T_{19}^{20} + 70 T_{19}^{19} + 70 T_{19}^{18} + \cdots + 1 \) acting on \(S_{1}^{\mathrm{new}}(3549, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{24} \) Copy content Toggle raw display
$3$ \( T^{24} - T^{22} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( T^{24} \) Copy content Toggle raw display
$7$ \( T^{24} - T^{22} + \cdots + 1 \) Copy content Toggle raw display
$11$ \( T^{24} \) Copy content Toggle raw display
$13$ \( (T^{2} + 1)^{12} \) Copy content Toggle raw display
$17$ \( T^{24} \) Copy content Toggle raw display
$19$ \( T^{24} + 2 T^{23} + \cdots + 1 \) Copy content Toggle raw display
$23$ \( T^{24} \) Copy content Toggle raw display
$29$ \( T^{24} \) Copy content Toggle raw display
$31$ \( T^{24} + 2 T^{23} + \cdots + 1 \) Copy content Toggle raw display
$37$ \( T^{24} - 2 T^{23} + \cdots + 1 \) Copy content Toggle raw display
$41$ \( T^{24} \) Copy content Toggle raw display
$43$ \( (T^{12} + 13 T^{8} + \cdots + 13)^{2} \) Copy content Toggle raw display
$47$ \( T^{24} \) Copy content Toggle raw display
$53$ \( T^{24} \) Copy content Toggle raw display
$59$ \( T^{24} \) Copy content Toggle raw display
$61$ \( T^{24} - 4 T^{22} + \cdots + 1 \) Copy content Toggle raw display
$67$ \( T^{24} - 2 T^{23} + \cdots + 1 \) Copy content Toggle raw display
$71$ \( T^{24} \) Copy content Toggle raw display
$73$ \( T^{24} - 2 T^{23} + \cdots + 1 \) Copy content Toggle raw display
$79$ \( T^{24} + 65 T^{18} + \cdots + 169 \) Copy content Toggle raw display
$83$ \( T^{24} \) Copy content Toggle raw display
$89$ \( T^{24} \) Copy content Toggle raw display
$97$ \( T^{24} - 2 T^{23} + \cdots + 4096 \) Copy content Toggle raw display
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