Properties

Label 2151.1.d.e
Level $2151$
Weight $1$
Character orbit 2151.d
Self dual yes
Analytic conductor $1.073$
Analytic rank $0$
Dimension $4$
Projective image $D_{15}$
CM discriminant -239
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 2151 = 3^{2} \cdot 239 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2151.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: yes
Analytic conductor: \(1.07348884217\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{15})^+\)
Defining polynomial: \(x^{4} - x^{3} - 4 x^{2} + 4 x + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 239)
Projective image \(D_{15}\)
Projective field Galois closure of 15.1.44543599279432079.1

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta_{1} q^{2} + ( 1 + \beta_{2} ) q^{4} + ( -1 + \beta_{1} - \beta_{3} ) q^{5} + ( -\beta_{1} - \beta_{3} ) q^{8} +O(q^{10})\) \( q -\beta_{1} q^{2} + ( 1 + \beta_{2} ) q^{4} + ( -1 + \beta_{1} - \beta_{3} ) q^{5} + ( -\beta_{1} - \beta_{3} ) q^{8} + ( -1 + \beta_{3} ) q^{10} + ( \beta_{2} + \beta_{3} ) q^{11} + ( 2 - \beta_{1} + \beta_{2} + \beta_{3} ) q^{16} + q^{17} + ( \beta_{1} - \beta_{2} ) q^{20} + ( -1 - \beta_{2} - 2 \beta_{3} ) q^{22} + ( 1 - \beta_{2} - \beta_{3} ) q^{25} -\beta_{2} q^{29} + \beta_{1} q^{31} + ( 1 - \beta_{1} - \beta_{3} ) q^{32} -\beta_{1} q^{34} + ( -1 + \beta_{1} - \beta_{2} ) q^{40} + ( 2 + \beta_{2} + 2 \beta_{3} ) q^{44} + q^{49} + ( 1 - \beta_{1} + \beta_{2} + 2 \beta_{3} ) q^{50} + ( 1 - \beta_{1} + 2 \beta_{3} ) q^{55} + ( \beta_{1} + \beta_{3} ) q^{58} - q^{61} + ( -2 - \beta_{2} ) q^{62} + ( 1 - \beta_{1} + \beta_{2} ) q^{64} - q^{67} + ( 1 + \beta_{2} ) q^{68} -\beta_{3} q^{71} + ( -2 + \beta_{1} + \beta_{3} ) q^{80} -\beta_{2} q^{83} + ( -1 + \beta_{1} - \beta_{3} ) q^{85} + ( -1 - \beta_{1} - \beta_{2} - \beta_{3} ) q^{88} -\beta_{1} q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - q^{2} + 5q^{4} - q^{5} + q^{8} + O(q^{10}) \) \( 4q - q^{2} + 5q^{4} - q^{5} + q^{8} - 6q^{10} - q^{11} + 6q^{16} + 4q^{17} - q^{22} + 5q^{25} - q^{29} + q^{31} + 5q^{32} - q^{34} - 4q^{40} + 5q^{44} + 4q^{49} - q^{55} - q^{58} - 4q^{61} - 9q^{62} + 4q^{64} - 4q^{67} + 5q^{68} + 2q^{71} - 9q^{80} - q^{83} - q^{85} - 4q^{88} - q^{98} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2151\mathbb{Z}\right)^\times\).

\(n\) \(479\) \(1441\)
\(\chi(n)\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
955.1
1.82709
1.33826
−0.209057
−1.95630
−1.82709 0 2.33826 0.209057 0 0 −2.44512 0 −0.381966
955.2 −1.33826 0 0.790943 1.95630 0 0 0.279773 0 −2.61803
955.3 0.209057 0 −0.956295 −1.82709 0 0 −0.408977 0 −0.381966
955.4 1.95630 0 2.82709 −1.33826 0 0 3.57433 0 −2.61803
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
239.b odd 2 1 CM by \(\Q(\sqrt{-239}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2151.1.d.e 4
3.b odd 2 1 239.1.b.c 4
12.b even 2 1 3824.1.h.c 4
239.b odd 2 1 CM 2151.1.d.e 4
717.b even 2 1 239.1.b.c 4
2868.e odd 2 1 3824.1.h.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
239.1.b.c 4 3.b odd 2 1
239.1.b.c 4 717.b even 2 1
2151.1.d.e 4 1.a even 1 1 trivial
2151.1.d.e 4 239.b odd 2 1 CM
3824.1.h.c 4 12.b even 2 1
3824.1.h.c 4 2868.e odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(2151, [\chi])\):

\( T_{2}^{4} + T_{2}^{3} - 4 T_{2}^{2} - 4 T_{2} + 1 \)
\( T_{5}^{4} + T_{5}^{3} - 4 T_{5}^{2} - 4 T_{5} + 1 \)

Hecke characteristic polynomials

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