Properties

Label 1859.1.c.b
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: \( x^{3} - x^{2} - 2x + 1 \) Copy content Toggle raw display
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: $D_{14}$
Artin field: Galois closure of 14.2.536561726709678316933.1

$q$-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.

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

Basis of coefficient ring in terms of \(\nu = \zeta_{14} + \zeta_{14}^{-1}\):

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 2 \) Copy content Toggle raw display

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

Hecke kernels

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

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

Hecke characteristic polynomials

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