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Properties

Label 1183.1.n.b
Level $1183$
Weight $1$
Character orbit 1183.n
Analytic conductor $0.590$
Analytic rank $0$
Dimension $6$
Projective image $D_{7}$
CM discriminant -7
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1183,1,Mod(146,1183)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1183, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 2]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1183.146");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1183 = 7 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1183.n (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: \(0.590393909945\)
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: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{7}\)
Projective field: Galois closure of 7.1.1655595487.1

$q$-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.

\(f(q)\) \(=\) \( q + ( - \beta_{5} - \beta_{4} - \beta_1 + 1) q^{2} + ( - \beta_{5} - \beta_{4} - \beta_{3}) q^{4} - \beta_{5} q^{7} + ( - \beta_{3} - 1) q^{8} - \beta_{5} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{5} - \beta_{4} - \beta_1 + 1) q^{2} + ( - \beta_{5} - \beta_{4} - \beta_{3}) q^{4} - \beta_{5} q^{7} + ( - \beta_{3} - 1) q^{8} - \beta_{5} q^{9} + \beta_{4} q^{11} + ( - \beta_{3} + \beta_{2} - 1) q^{14} + (\beta_{5} + \beta_{4} + \beta_1 - 1) q^{16} + ( - \beta_{3} + \beta_{2} - 1) q^{18} + (\beta_{5} + \beta_{4} + \beta_{3}) q^{22} + \beta_1 q^{23} + q^{25} + (\beta_{5} + \beta_{4} - 1) q^{28} + ( - \beta_{5} - \beta_{4} - \beta_1 + 1) q^{29} + \beta_{5} q^{32} + (\beta_{5} + \beta_{4} - 1) q^{36} + \beta_1 q^{37} + (\beta_{5} + \beta_{4} + \beta_{3} + \cdots + \beta_1) q^{43}+ \cdots + \beta_{3} q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + q^{2} - 2 q^{4} - 3 q^{7} - 4 q^{8} - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + q^{2} - 2 q^{4} - 3 q^{7} - 4 q^{8} - 3 q^{9} + q^{11} - 2 q^{14} - q^{16} - 2 q^{18} + 2 q^{22} + q^{23} + 6 q^{25} - 2 q^{28} + q^{29} + 3 q^{32} - 2 q^{36} + q^{37} + q^{43} + 8 q^{44} + 2 q^{46} - 3 q^{49} + q^{50} - 2 q^{53} + 2 q^{56} - 5 q^{58} - 3 q^{63} + q^{67} + q^{71} + 2 q^{72} + 2 q^{74} - 2 q^{77} - 2 q^{79} - 3 q^{81} + 10 q^{86} + 4 q^{88} - 6 q^{92} + q^{98} - 2 q^{99}+O(q^{100}) \) 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 \(\left(\mathbb{Z}/1183\mathbb{Z}\right)^\times\).

\(n\) \(339\) \(1016\)
\(\chi(n)\) \(-1\) \(-\beta_{5}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
146.1
0.900969 + 1.56052i
−0.623490 − 1.07992i
0.222521 + 0.385418i
0.900969 − 1.56052i
−0.623490 + 1.07992i
0.222521 − 0.385418i
−0.623490 − 1.07992i 0 −0.277479 + 0.480608i 0 0 −0.500000 + 0.866025i −0.554958 −0.500000 + 0.866025i 0
146.2 0.222521 + 0.385418i 0 0.400969 − 0.694498i 0 0 −0.500000 + 0.866025i 0.801938 −0.500000 + 0.866025i 0
146.3 0.900969 + 1.56052i 0 −1.12349 + 1.94594i 0 0 −0.500000 + 0.866025i −2.24698 −0.500000 + 0.866025i 0
867.1 −0.623490 + 1.07992i 0 −0.277479 − 0.480608i 0 0 −0.500000 − 0.866025i −0.554958 −0.500000 − 0.866025i 0
867.2 0.222521 − 0.385418i 0 0.400969 + 0.694498i 0 0 −0.500000 − 0.866025i 0.801938 −0.500000 − 0.866025i 0
867.3 0.900969 − 1.56052i 0 −1.12349 − 1.94594i 0 0 −0.500000 − 0.866025i −2.24698 −0.500000 − 0.866025i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 146.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 CM by \(\Q(\sqrt{-7}) \)
13.c even 3 1 inner
91.n odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1183.1.n.b 6
7.b odd 2 1 CM 1183.1.n.b 6
13.b even 2 1 1183.1.n.a 6
13.c even 3 1 1183.1.d.a ✓ 3
13.c even 3 1 inner 1183.1.n.b 6
13.d odd 4 2 1183.1.t.a 12
13.e even 6 1 1183.1.d.b yes 3
13.e even 6 1 1183.1.n.a 6
13.f odd 12 2 1183.1.b.a 6
13.f odd 12 2 1183.1.t.a 12
91.b odd 2 1 1183.1.n.a 6
91.i even 4 2 1183.1.t.a 12
91.n odd 6 1 1183.1.d.a ✓ 3
91.n odd 6 1 inner 1183.1.n.b 6
91.t odd 6 1 1183.1.d.b yes 3
91.t odd 6 1 1183.1.n.a 6
91.bc even 12 2 1183.1.b.a 6
91.bc even 12 2 1183.1.t.a 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1183.1.b.a 6 13.f odd 12 2
1183.1.b.a 6 91.bc even 12 2
1183.1.d.a ✓ 3 13.c even 3 1
1183.1.d.a ✓ 3 91.n odd 6 1
1183.1.d.b yes 3 13.e even 6 1
1183.1.d.b yes 3 91.t odd 6 1
1183.1.n.a 6 13.b even 2 1
1183.1.n.a 6 13.e even 6 1
1183.1.n.a 6 91.b odd 2 1
1183.1.n.a 6 91.t odd 6 1
1183.1.n.b 6 1.a even 1 1 trivial
1183.1.n.b 6 7.b odd 2 1 CM
1183.1.n.b 6 13.c even 3 1 inner
1183.1.n.b 6 91.n odd 6 1 inner
1183.1.t.a 12 13.d odd 4 2
1183.1.t.a 12 13.f odd 12 2
1183.1.t.a 12 91.i even 4 2
1183.1.t.a 12 91.bc even 12 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{6} - T_{2}^{5} + 3T_{2}^{4} + 5T_{2}^{2} - 2T_{2} + 1 \) acting on \(S_{1}^{\mathrm{new}}(1183, [\chi])\). 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
$3$ \( T^{6} \) Copy content Toggle raw display
$5$ \( T^{6} \) Copy content Toggle raw display
$7$ \( (T^{2} + T + 1)^{3} \) Copy content Toggle raw display
$11$ \( T^{6} - T^{5} + 3 T^{4} + \cdots + 1 \) Copy content Toggle raw display
$13$ \( T^{6} \) Copy content Toggle raw display
$17$ \( T^{6} \) Copy content Toggle raw display
$19$ \( T^{6} \) Copy content Toggle raw display
$23$ \( T^{6} - T^{5} + 3 T^{4} + \cdots + 1 \) Copy content Toggle raw display
$29$ \( T^{6} - T^{5} + 3 T^{4} + \cdots + 1 \) 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
$41$ \( T^{6} \) Copy content Toggle raw display
$43$ \( T^{6} - T^{5} + 3 T^{4} + \cdots + 1 \) Copy content Toggle raw display
$47$ \( T^{6} \) Copy content Toggle raw display
$53$ \( (T^{3} + T^{2} - 2 T - 1)^{2} \) Copy content Toggle raw display
$59$ \( T^{6} \) Copy content Toggle raw display
$61$ \( T^{6} \) Copy content Toggle raw display
$67$ \( T^{6} - T^{5} + 3 T^{4} + \cdots + 1 \) Copy content Toggle raw display
$71$ \( T^{6} - T^{5} + 3 T^{4} + \cdots + 1 \) Copy content Toggle raw display
$73$ \( T^{6} \) Copy content Toggle raw display
$79$ \( (T^{3} + T^{2} - 2 T - 1)^{2} \) Copy content Toggle raw display
$83$ \( T^{6} \) Copy content Toggle raw display
$89$ \( T^{6} \) Copy content Toggle raw display
$97$ \( T^{6} \) Copy content Toggle raw display
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