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Properties

Label 1859.1.k.a
Level $1859$
Weight $1$
Character orbit 1859.k
Analytic conductor $0.928$
Analytic rank $0$
Dimension $6$
Projective image $D_{7}$
CM discriminant -11
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1859,1,Mod(1374,1859)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1859, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 4]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1859.1374");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1859 = 11 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1859.k (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: \(0.927761858485\)
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 7.1.6424482779.1
Artin image: $C_3\times D_7$
Artin field: Galois closure of \(\mathbb{Q}[x]/(x^{21} - \cdots)\)

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{4} q^{3} - \beta_{5} q^{4} - \beta_{2} q^{5} + ( - \beta_{5} + \beta_{2} - \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{4} q^{3} - \beta_{5} q^{4} - \beta_{2} q^{5} + ( - \beta_{5} + \beta_{2} - \beta_1) q^{9} + (\beta_{5} - 1) q^{11} + \beta_{3} q^{12} + ( - \beta_{5} - \beta_1 + 1) q^{15} + (\beta_{5} - 1) q^{16} + (\beta_{2} - \beta_1) q^{20} + ( - \beta_{5} - \beta_{4} - \beta_1 + 1) q^{23} + ( - \beta_{3} + \beta_{2}) q^{25} + (\beta_{2} - 1) q^{27} + ( - \beta_{3} + \beta_{2} - 1) q^{31} + ( - \beta_{4} - \beta_{3}) q^{33} + (\beta_{5} + \beta_1 - 1) q^{36} + \beta_{4} q^{37} + q^{44} + ( - \beta_{5} + \beta_{4} + \beta_{3}) q^{45} + ( - \beta_{3} + \beta_{2} - 1) q^{47} + ( - \beta_{4} - \beta_{3}) q^{48} + (\beta_{5} - 1) q^{49} - \beta_{2} q^{53} + \beta_1 q^{55} + ( - \beta_{4} - \beta_{3}) q^{59} + (\beta_{2} - 1) q^{60} + q^{64} + (2 \beta_{5} - 2) q^{67} + (\beta_{5} + \beta_{4} + \beta_{3}) q^{69} + ( - \beta_{4} - \beta_{3}) q^{71} + ( - \beta_{5} + 1) q^{75} + \beta_1 q^{80} - \beta_{4} q^{81} + ( - \beta_{5} - \beta_{4} - \beta_1 + 1) q^{89} + ( - \beta_{3} + \beta_{2} - 1) q^{92} + ( - \beta_{5} - \beta_{4} + 1) q^{93} + (\beta_{2} - \beta_1) q^{97} + ( - \beta_{2} + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + q^{3} - 3 q^{4} - 2 q^{5} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + q^{3} - 3 q^{4} - 2 q^{5} - 2 q^{9} - 3 q^{11} - 2 q^{12} + 2 q^{15} - 3 q^{16} + q^{20} + q^{23} + 4 q^{25} - 4 q^{27} - 2 q^{31} + q^{33} - 2 q^{36} + q^{37} + 6 q^{44} - 4 q^{45} - 2 q^{47} + q^{48} - 3 q^{49} - 2 q^{53} + q^{55} + q^{59} - 4 q^{60} + 6 q^{64} - 6 q^{67} + 2 q^{69} + q^{71} + 3 q^{75} + q^{80} - q^{81} + q^{89} - 2 q^{92} + 2 q^{93} + q^{97} + 4 q^{99}+O(q^{100}) \) 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 \(\left(\mathbb{Z}/1859\mathbb{Z}\right)^\times\).

\(n\) \(508\) \(1354\)
\(\chi(n)\) \(-1\) \(-\beta_{5}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1374.1
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 −0.623490 + 1.07992i −0.500000 − 0.866025i −0.445042 0 0 0 −0.277479 − 0.480608i 0
1374.2 0 0.222521 − 0.385418i −0.500000 − 0.866025i −1.80194 0 0 0 0.400969 + 0.694498i 0
1374.3 0 0.900969 − 1.56052i −0.500000 − 0.866025i 1.24698 0 0 0 −1.12349 − 1.94594i 0
1836.1 0 −0.623490 − 1.07992i −0.500000 + 0.866025i −0.445042 0 0 0 −0.277479 + 0.480608i 0
1836.2 0 0.222521 + 0.385418i −0.500000 + 0.866025i −1.80194 0 0 0 0.400969 − 0.694498i 0
1836.3 0 0.900969 + 1.56052i −0.500000 + 0.866025i 1.24698 0 0 0 −1.12349 + 1.94594i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1374.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.b odd 2 1 CM by \(\Q(\sqrt{-11}) \)
13.c even 3 1 inner
143.k odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1859.1.k.a 6
11.b odd 2 1 CM 1859.1.k.a 6
13.b even 2 1 1859.1.k.b 6
13.c even 3 1 1859.1.c.a ✓ 3
13.c even 3 1 inner 1859.1.k.a 6
13.d odd 4 2 1859.1.i.c 12
13.e even 6 1 1859.1.c.b yes 3
13.e even 6 1 1859.1.k.b 6
13.f odd 12 2 1859.1.d.a 6
13.f odd 12 2 1859.1.i.c 12
143.d odd 2 1 1859.1.k.b 6
143.g even 4 2 1859.1.i.c 12
143.i odd 6 1 1859.1.c.b yes 3
143.i odd 6 1 1859.1.k.b 6
143.k odd 6 1 1859.1.c.a ✓ 3
143.k odd 6 1 inner 1859.1.k.a 6
143.o even 12 2 1859.1.d.a 6
143.o even 12 2 1859.1.i.c 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1859.1.c.a ✓ 3 13.c even 3 1
1859.1.c.a ✓ 3 143.k odd 6 1
1859.1.c.b yes 3 13.e even 6 1
1859.1.c.b yes 3 143.i odd 6 1
1859.1.d.a 6 13.f odd 12 2
1859.1.d.a 6 143.o even 12 2
1859.1.i.c 12 13.d odd 4 2
1859.1.i.c 12 13.f odd 12 2
1859.1.i.c 12 143.g even 4 2
1859.1.i.c 12 143.o even 12 2
1859.1.k.a 6 1.a even 1 1 trivial
1859.1.k.a 6 11.b odd 2 1 CM
1859.1.k.a 6 13.c even 3 1 inner
1859.1.k.a 6 143.k odd 6 1 inner
1859.1.k.b 6 13.b even 2 1
1859.1.k.b 6 13.e even 6 1
1859.1.k.b 6 143.d odd 2 1
1859.1.k.b 6 143.i odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(1859, [\chi])\):

\( T_{2} \) Copy content Toggle raw display
\( T_{5}^{3} + T_{5}^{2} - 2T_{5} - 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( T^{6} - T^{5} + 3 T^{4} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( (T^{3} + T^{2} - 2 T - 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{6} \) Copy content Toggle raw display
$11$ \( (T^{2} + T + 1)^{3} \) 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
$29$ \( T^{6} \) Copy content Toggle raw display
$31$ \( (T^{3} + T^{2} - 2 T - 1)^{2} \) 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
$43$ \( T^{6} \) Copy content Toggle raw display
$47$ \( (T^{3} + T^{2} - 2 T - 1)^{2} \) Copy content Toggle raw display
$53$ \( (T^{3} + T^{2} - 2 T - 1)^{2} \) Copy content Toggle raw display
$59$ \( T^{6} - T^{5} + 3 T^{4} + \cdots + 1 \) Copy content Toggle raw display
$61$ \( T^{6} \) Copy content Toggle raw display
$67$ \( (T^{2} + 2 T + 4)^{3} \) 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
$79$ \( T^{6} \) 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
$97$ \( T^{6} - T^{5} + 3 T^{4} + \cdots + 1 \) Copy content Toggle raw display
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