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Properties

Label 2009.1.i.a
Level $2009$
Weight $1$
Character orbit 2009.i
Analytic conductor $1.003$
Analytic rank $0$
Dimension $6$
Projective image $D_{7}$
CM discriminant -287
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2009,1,Mod(901,2009)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2009, 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("2009.901");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2009 = 7^{2} \cdot 41 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2009.i (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: \(1.00262161038\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{6})\)
Coefficient field: 6.0.64827.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{5} + 3x^{4} + 5x^{2} - 2x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 287)
Projective image: \(D_{7}\)
Projective field: Galois closure of 7.1.23639903.1
Artin image: $C_3\times D_{14}$
Artin field: Galois closure of \(\mathbb{Q}[x]/(x^{42} + \cdots)\)

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + ( - \beta_{5} - \beta_{4} + \cdots - \beta_1) q^{3}+ \cdots + (\beta_{5} + \beta_{4} - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + ( - \beta_{5} - \beta_{4} + \cdots - \beta_1) q^{3}+ \cdots + (\beta_{3} - \beta_{2} + 1) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + q^{2} - q^{3} - 2 q^{4} + 4 q^{6} - 4 q^{8} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + q^{2} - q^{3} - 2 q^{4} + 4 q^{6} - 4 q^{8} - 2 q^{9} + 4 q^{12} + 2 q^{13} - q^{16} - q^{17} + 3 q^{18} - q^{19} + q^{23} - 4 q^{24} - 3 q^{25} + 5 q^{26} + 4 q^{27} + 3 q^{32} + 4 q^{34} - 2 q^{36} + q^{37} - 2 q^{38} + 2 q^{39} - 6 q^{41} - 2 q^{43} + 2 q^{46} - q^{47} - 4 q^{48} - 2 q^{50} + 2 q^{51} - 3 q^{52} + 3 q^{54} - 4 q^{57} - 3 q^{68} - 10 q^{69} - q^{72} - 5 q^{74} - q^{75} + 6 q^{76} + 6 q^{78} - q^{81} - q^{82} + 2 q^{86} - q^{89} + 8 q^{92} + 5 q^{94} + q^{96} + 2 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - x^{5} + 3x^{4} + 5x^{2} - 2x + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{5} + 3\nu^{4} - 9\nu^{3} + 5\nu^{2} - 2\nu + 6 ) / 13 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -3\nu^{5} + 9\nu^{4} - 14\nu^{3} + 15\nu^{2} - 6\nu + 18 ) / 13 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -4\nu^{5} - \nu^{4} - 10\nu^{3} - 6\nu^{2} - 34\nu - 2 ) / 13 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -6\nu^{5} + 5\nu^{4} - 15\nu^{3} - 9\nu^{2} - 25\nu + 10 ) / 13 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{5} + \beta_{4} + \beta_{3} - \beta_{2} + \beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} - 3\beta_{2} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2\beta_{5} - 3\beta_{4} - 4\beta _1 - 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{5} - 4\beta_{4} - 4\beta_{3} + 9\beta_{2} - 9\beta_1 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2009\mathbb{Z}\right)^\times\).

\(n\) \(493\) \(785\)
\(\chi(n)\) \(1 - \beta_{5}\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
901.1
−0.623490 + 1.07992i
0.222521 − 0.385418i
0.900969 − 1.56052i
−0.623490 − 1.07992i
0.222521 + 0.385418i
0.900969 + 1.56052i
−0.623490 + 1.07992i −0.222521 − 0.385418i −0.277479 − 0.480608i 0 0.554958 0 −0.554958 0.400969 − 0.694498i 0
901.2 0.222521 − 0.385418i −0.900969 − 1.56052i 0.400969 + 0.694498i 0 −0.801938 0 0.801938 −1.12349 + 1.94594i 0
901.3 0.900969 − 1.56052i 0.623490 + 1.07992i −1.12349 − 1.94594i 0 2.24698 0 −2.24698 −0.277479 + 0.480608i 0
1844.1 −0.623490 − 1.07992i −0.222521 + 0.385418i −0.277479 + 0.480608i 0 0.554958 0 −0.554958 0.400969 + 0.694498i 0
1844.2 0.222521 + 0.385418i −0.900969 + 1.56052i 0.400969 − 0.694498i 0 −0.801938 0 0.801938 −1.12349 − 1.94594i 0
1844.3 0.900969 + 1.56052i 0.623490 − 1.07992i −1.12349 + 1.94594i 0 2.24698 0 −2.24698 −0.277479 − 0.480608i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 901.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
287.d odd 2 1 CM by \(\Q(\sqrt{-287}) \)
7.c even 3 1 inner
287.i odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2009.1.i.a 6
7.b odd 2 1 2009.1.i.b 6
7.c even 3 1 287.1.d.b yes 3
7.c even 3 1 inner 2009.1.i.a 6
7.d odd 6 1 287.1.d.a ✓ 3
7.d odd 6 1 2009.1.i.b 6
21.g even 6 1 2583.1.f.b 3
21.h odd 6 1 2583.1.f.a 3
41.b even 2 1 2009.1.i.b 6
287.d odd 2 1 CM 2009.1.i.a 6
287.i odd 6 1 287.1.d.b yes 3
287.i odd 6 1 inner 2009.1.i.a 6
287.j even 6 1 287.1.d.a ✓ 3
287.j even 6 1 2009.1.i.b 6
861.r even 6 1 2583.1.f.a 3
861.t odd 6 1 2583.1.f.b 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
287.1.d.a ✓ 3 7.d odd 6 1
287.1.d.a ✓ 3 287.j even 6 1
287.1.d.b yes 3 7.c even 3 1
287.1.d.b yes 3 287.i odd 6 1
2009.1.i.a 6 1.a even 1 1 trivial
2009.1.i.a 6 7.c even 3 1 inner
2009.1.i.a 6 287.d odd 2 1 CM
2009.1.i.a 6 287.i odd 6 1 inner
2009.1.i.b 6 7.b odd 2 1
2009.1.i.b 6 7.d odd 6 1
2009.1.i.b 6 41.b even 2 1
2009.1.i.b 6 287.j even 6 1
2583.1.f.a 3 21.h odd 6 1
2583.1.f.a 3 861.r even 6 1
2583.1.f.b 3 21.g even 6 1
2583.1.f.b 3 861.t odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{6} + T_{3}^{5} + 3T_{3}^{4} + 5T_{3}^{2} + 2T_{3} + 1 \) acting on \(S_{1}^{\mathrm{new}}(2009, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

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