Properties

Label 1859.1.k.b
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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Newspace parameters

Level: \( N \) \(=\) \( 1859 = 11 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1859.k (of order \(6\), degree \(2\), not minimal)

Newform invariants

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
Defining polynomial: \(x^{6} - x^{5} + 3 x^{4} + 5 x^{2} - 2 x + 1\)
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: $C_6\times D_7$
Artin field: Galois closure of \(\mathbb{Q}[x]/(x^{42} - \cdots)\)

$q$-expansion

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.

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

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

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{5} + 3 \nu^{4} - 9 \nu^{3} + 5 \nu^{2} - 2 \nu + 6 \)\()/13\)
\(\beta_{3}\)\(=\)\((\)\( -3 \nu^{5} + 9 \nu^{4} - 14 \nu^{3} + 15 \nu^{2} - 6 \nu + 18 \)\()/13\)
\(\beta_{4}\)\(=\)\((\)\( -4 \nu^{5} - \nu^{4} - 10 \nu^{3} - 6 \nu^{2} - 34 \nu - 2 \)\()/13\)
\(\beta_{5}\)\(=\)\((\)\( -6 \nu^{5} + 5 \nu^{4} - 15 \nu^{3} - 9 \nu^{2} - 25 \nu + 10 \)\()/13\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(-\beta_{5} + \beta_{4} + \beta_{3} - \beta_{2} + \beta_{1}\)
\(\nu^{3}\)\(=\)\(\beta_{3} - 3 \beta_{2}\)
\(\nu^{4}\)\(=\)\(2 \beta_{5} - 3 \beta_{4} - 4 \beta_{1} - 2\)
\(\nu^{5}\)\(=\)\(\beta_{5} - 4 \beta_{4} - 4 \beta_{3} + 9 \beta_{2} - 9 \beta_{1}\)

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.

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 1836.3
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.b 6
11.b odd 2 1 CM 1859.1.k.b 6
13.b even 2 1 1859.1.k.a 6
13.c even 3 1 1859.1.c.b yes 3
13.c even 3 1 inner 1859.1.k.b 6
13.d odd 4 2 1859.1.i.c 12
13.e even 6 1 1859.1.c.a 3
13.e even 6 1 1859.1.k.a 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.a 6
143.g even 4 2 1859.1.i.c 12
143.i odd 6 1 1859.1.c.a 3
143.i odd 6 1 1859.1.k.a 6
143.k odd 6 1 1859.1.c.b yes 3
143.k odd 6 1 inner 1859.1.k.b 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.e even 6 1
1859.1.c.a 3 143.i odd 6 1
1859.1.c.b yes 3 13.c even 3 1
1859.1.c.b yes 3 143.k 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 13.b even 2 1
1859.1.k.a 6 13.e even 6 1
1859.1.k.a 6 143.d odd 2 1
1859.1.k.a 6 143.i odd 6 1
1859.1.k.b 6 1.a even 1 1 trivial
1859.1.k.b 6 11.b odd 2 1 CM
1859.1.k.b 6 13.c even 3 1 inner
1859.1.k.b 6 143.k odd 6 1 inner

Hecke kernels

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

\( T_{2} \)
\( T_{5}^{3} - T_{5}^{2} - 2 T_{5} + 1 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \)
$3$ \( 1 - 2 T + 5 T^{2} + 3 T^{4} - T^{5} + T^{6} \)
$5$ \( ( 1 - 2 T - T^{2} + T^{3} )^{2} \)
$7$ \( T^{6} \)
$11$ \( ( 1 - T + T^{2} )^{3} \)
$13$ \( T^{6} \)
$17$ \( T^{6} \)
$19$ \( T^{6} \)
$23$ \( 1 - 2 T + 5 T^{2} + 3 T^{4} - T^{5} + T^{6} \)
$29$ \( T^{6} \)
$31$ \( ( 1 - 2 T - T^{2} + T^{3} )^{2} \)
$37$ \( 1 + 2 T + 5 T^{2} + 3 T^{4} + T^{5} + T^{6} \)
$41$ \( T^{6} \)
$43$ \( T^{6} \)
$47$ \( ( 1 - 2 T - T^{2} + T^{3} )^{2} \)
$53$ \( ( -1 - 2 T + T^{2} + T^{3} )^{2} \)
$59$ \( 1 + 2 T + 5 T^{2} + 3 T^{4} + T^{5} + T^{6} \)
$61$ \( T^{6} \)
$67$ \( ( 4 - 2 T + T^{2} )^{3} \)
$71$ \( 1 + 2 T + 5 T^{2} + 3 T^{4} + T^{5} + T^{6} \)
$73$ \( T^{6} \)
$79$ \( T^{6} \)
$83$ \( T^{6} \)
$89$ \( 1 + 2 T + 5 T^{2} + 3 T^{4} + T^{5} + T^{6} \)
$97$ \( 1 + 2 T + 5 T^{2} + 3 T^{4} + T^{5} + T^{6} \)
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