Properties

Label 2151.1.d.c
Level $2151$
Weight $1$
Character orbit 2151.d
Self dual yes
Analytic conductor $1.073$
Analytic rank $0$
Dimension $2$
Projective image $D_{5}$
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: \(2\)
Coefficient field: \(\Q(\sqrt{5}) \)
Defining polynomial: \(x^{2} - x - 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 239)
Projective image \(D_{5}\)
Projective field Galois closure of 5.1.57121.1
Artin image $D_{10}$
Artin field Galois closure of 10.0.792862499763.1

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{5})\). We also show the integral \(q\)-expansion of the trace form.

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

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
1.61803
−0.618034
−0.618034 0 −0.618034 −0.618034 0 0 1.00000 0 0.381966
955.2 1.61803 0 1.61803 1.61803 0 0 1.00000 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.c 2
3.b odd 2 1 239.1.b.b 2
12.b even 2 1 3824.1.h.b 2
239.b odd 2 1 CM 2151.1.d.c 2
717.b even 2 1 239.1.b.b 2
2868.e odd 2 1 3824.1.h.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
239.1.b.b 2 3.b odd 2 1
239.1.b.b 2 717.b even 2 1
2151.1.d.c 2 1.a even 1 1 trivial
2151.1.d.c 2 239.b odd 2 1 CM
3824.1.h.b 2 12.b even 2 1
3824.1.h.b 2 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 \(S_{1}^{\mathrm{new}}(2151, [\chi])\):

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

Hecke characteristic polynomials

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