Properties

Label 1859.1.c.a
Level $1859$
Weight $1$
Character orbit 1859.c
Self dual yes
Analytic conductor $0.928$
Analytic rank $0$
Dimension $3$
Projective image $D_{7}$
CM discriminant -11
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1859 = 11 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1859.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: yes
Analytic conductor: \(0.927761858485\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: \(\Q(\zeta_{14})^+\)
Defining polynomial: \(x^{3} - x^{2} - 2 x + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{7}\)
Projective field: Galois closure of 7.1.6424482779.1
Artin image: $D_7$
Artin field: Galois closure of 7.1.6424482779.1

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1859\mathbb{Z}\right)^\times\).

\(n\) \(508\) \(1354\)
\(\chi(n)\) \(-1\) \(1\)

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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
846.1
−1.24698
1.80194
0.445042
0 −1.80194 1.00000 1.24698 0 0 0 2.24698 0
846.2 0 −0.445042 1.00000 −1.80194 0 0 0 −0.801938 0
846.3 0 1.24698 1.00000 −0.445042 0 0 0 0.554958 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.b odd 2 1 CM by \(\Q(\sqrt{-11}) \)

Twists

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

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

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