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Properties

Label 2007.1.v.a
Level $2007$
Weight $1$
Character orbit 2007.v
Analytic conductor $1.002$
Analytic rank $0$
Dimension $36$
Projective image $D_{74}$
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] = [2007,1,Mod(91,2007)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2007, base_ring=CyclotomicField(74))
 
chi = DirichletCharacter(H, H._module([0, 45]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2007.91");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2007 = 3^{2} \cdot 223 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2007.v (of order \(74\), degree \(36\), 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: \(1.00162348035\)
Analytic rank: \(0\)
Dimension: \(36\)
Coefficient field: \(\Q(\zeta_{74})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{36} - x^{35} + x^{34} - x^{33} + x^{32} - x^{31} + x^{30} - x^{29} + x^{28} - x^{27} + x^{26} + \cdots + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{74}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{74} - \cdots)\)

$q$-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.

\(f(q)\) \(=\) \( q - \zeta_{74}^{20} q^{4} + (\zeta_{74}^{27} + \zeta_{74}^{21}) q^{7} +O(q^{10}) \) Copy content Toggle raw display \( q - \zeta_{74}^{20} q^{4} + (\zeta_{74}^{27} + \zeta_{74}^{21}) q^{7} + (\zeta_{74}^{32} + \zeta_{74}^{9}) q^{13} - \zeta_{74}^{3} q^{16} + (\zeta_{74}^{28} + \zeta_{74}^{24}) q^{19} + \zeta_{74}^{14} q^{25} + (\zeta_{74}^{10} + \zeta_{74}^{4}) q^{28} + ( - \zeta_{74}^{36} - \zeta_{74}^{6}) q^{31} + ( - \zeta_{74}^{13} + \zeta_{74}^{8}) q^{37} + (\zeta_{74}^{34} + \zeta_{74}^{12}) q^{43} + ( - \zeta_{74}^{17} + \cdots - \zeta_{74}^{5}) q^{49} + \cdots + ( - \zeta_{74}^{31} + \zeta_{74}^{29}) q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 36 q + q^{4} + 2 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 36 q + q^{4} + 2 q^{7} - q^{16} - 2 q^{19} - q^{25} - 2 q^{28} + 2 q^{31} - 2 q^{37} - 2 q^{43} - 3 q^{49} + q^{64} + 2 q^{73} + 2 q^{76}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2007\mathbb{Z}\right)^\times\).

\(n\) \(226\) \(893\)
\(\chi(n)\) \(\zeta_{74}^{33}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
91.1
−0.985616 + 0.169001i
−0.524307 − 0.851529i
−0.0424412 − 0.999099i
0.721956 − 0.691939i
0.594633 + 0.803997i
0.292823 + 0.956167i
−0.985616 − 0.169001i
0.828510 + 0.559975i
−0.660675 + 0.750672i
−0.873014 − 0.487695i
0.594633 − 0.803997i
0.828510 − 0.559975i
0.292823 − 0.956167i
−0.942877 − 0.333140i
0.721956 + 0.691939i
−0.873014 + 0.487695i
−0.210679 − 0.977555i
−0.660675 − 0.750672i
−0.372856 − 0.927889i
0.996397 + 0.0848059i
0 0 0.967733 − 0.251978i 0 0 1.03825 − 1.40380i 0 0 0
118.1 0 0 −0.0424412 − 0.999099i 0 0 1.55047 − 1.25191i 0 0 0
163.1 0 0 −0.660675 + 0.750672i 0 0 0.133193 − 0.216319i 0 0 0
190.1 0 0 0.911228 + 0.411901i 0 0 −1.15356 − 0.644415i 0 0 0
208.1 0 0 −0.985616 + 0.169001i 0 0 1.71835 + 0.776745i 0 0 0
334.1 0 0 −0.942877 − 0.333140i 0 0 −1.02806 + 1.16810i 0 0 0
397.1 0 0 0.967733 + 0.251978i 0 0 1.03825 + 1.40380i 0 0 0
442.1 0 0 −0.778036 + 0.628220i 0 0 0.0535200 − 0.417946i 0 0 0
505.1 0 0 0.292823 − 0.956167i 0 0 0.0703259 − 1.65553i 0 0 0
541.1 0 0 0.721956 + 0.691939i 0 0 −0.0800337 + 0.0282777i 0 0 0
550.1 0 0 −0.985616 − 0.169001i 0 0 1.71835 − 0.776745i 0 0 0
613.1 0 0 −0.778036 − 0.628220i 0 0 0.0535200 + 0.417946i 0 0 0
667.1 0 0 −0.942877 + 0.333140i 0 0 −1.02806 − 1.16810i 0 0 0
721.1 0 0 −0.873014 − 0.487695i 0 0 0.307058 − 1.00265i 0 0 0
919.1 0 0 0.911228 − 0.411901i 0 0 −1.15356 + 0.644415i 0 0 0
946.1 0 0 0.721956 − 0.691939i 0 0 −0.0800337 − 0.0282777i 0 0 0
1000.1 0 0 0.450204 − 0.892926i 0 0 0.443426 + 1.10351i 0 0 0
1081.1 0 0 0.292823 + 0.956167i 0 0 0.0703259 + 1.65553i 0 0 0
1099.1 0 0 −0.210679 + 0.977555i 0 0 −1.76365 − 0.459219i 0 0 0
1108.1 0 0 0.127018 − 0.991900i 0 0 −0.871354 + 1.72823i 0 0 0
See all 36 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 91.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)
223.f odd 74 1 inner
669.k even 74 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2007.1.v.a ✓ 36
3.b odd 2 1 CM 2007.1.v.a ✓ 36
223.f odd 74 1 inner 2007.1.v.a ✓ 36
669.k even 74 1 inner 2007.1.v.a ✓ 36
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2007.1.v.a ✓ 36 1.a even 1 1 trivial
2007.1.v.a ✓ 36 3.b odd 2 1 CM
2007.1.v.a ✓ 36 223.f odd 74 1 inner
2007.1.v.a ✓ 36 669.k even 74 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(2007, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{36} \) Copy content Toggle raw display
$3$ \( T^{36} \) Copy content Toggle raw display
$5$ \( T^{36} \) Copy content Toggle raw display
$7$ \( T^{36} - 2 T^{35} + \cdots + 1 \) Copy content Toggle raw display
$11$ \( T^{36} \) Copy content Toggle raw display
$13$ \( T^{36} - 222 T^{23} + \cdots + 37 \) Copy content Toggle raw display
$17$ \( T^{36} \) Copy content Toggle raw display
$19$ \( T^{36} + 2 T^{35} + \cdots + 1 \) Copy content Toggle raw display
$23$ \( T^{36} \) Copy content Toggle raw display
$29$ \( T^{36} \) Copy content Toggle raw display
$31$ \( T^{36} - 2 T^{35} + \cdots + 1 \) Copy content Toggle raw display
$37$ \( T^{36} + 2 T^{35} + \cdots + 1 \) Copy content Toggle raw display
$41$ \( T^{36} \) Copy content Toggle raw display
$43$ \( T^{36} + 2 T^{35} + \cdots + 1 \) Copy content Toggle raw display
$47$ \( T^{36} \) Copy content Toggle raw display
$53$ \( T^{36} \) Copy content Toggle raw display
$59$ \( T^{36} \) Copy content Toggle raw display
$61$ \( T^{36} + 37 T^{17} + \cdots + 37 \) Copy content Toggle raw display
$67$ \( T^{36} + 37 T^{17} + \cdots + 37 \) Copy content Toggle raw display
$71$ \( T^{36} \) Copy content Toggle raw display
$73$ \( T^{36} - 2 T^{35} + \cdots + 1 \) Copy content Toggle raw display
$79$ \( T^{36} - 37 T^{31} + \cdots + 37 \) Copy content Toggle raw display
$83$ \( T^{36} \) Copy content Toggle raw display
$89$ \( T^{36} \) Copy content Toggle raw display
$97$ \( T^{36} + 444 T^{26} + \cdots + 37 \) Copy content Toggle raw display
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