Properties

Label 2151.1.d.d
Level $2151$
Weight $1$
Character orbit 2151.d
Analytic conductor $1.073$
Analytic rank $0$
Dimension $2$
Projective image $S_{4}$
CM/RM no
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: no
Analytic conductor: \(1.07348884217\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-2}) \)
Defining polynomial: \(x^{2} + 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image \(S_{4}\)
Projective field Galois closure of 4.2.2151.1
Artin image $\GL(2,3)$
Artin field Galois closure of 8.2.9952248951.2

$q$-expansion

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

\(f(q)\) \(=\) \( q + q^{2} - q^{5} -\beta q^{7} - q^{8} +O(q^{10})\) \( q + q^{2} - q^{5} -\beta q^{7} - q^{8} - q^{10} - q^{11} + \beta q^{13} -\beta q^{14} - q^{16} - q^{17} + \beta q^{19} - q^{22} -\beta q^{23} + \beta q^{26} - q^{29} - q^{31} - q^{34} + \beta q^{35} -\beta q^{37} + \beta q^{38} + q^{40} + \beta q^{41} -\beta q^{46} -\beta q^{47} - q^{49} + q^{55} + \beta q^{56} - q^{58} + q^{61} - q^{62} + q^{64} -\beta q^{65} - q^{67} + \beta q^{70} -\beta q^{74} + \beta q^{77} + q^{80} + \beta q^{82} + q^{83} + q^{85} + q^{88} + 2 q^{91} -\beta q^{94} -\beta q^{95} - q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{2} - 2q^{5} - 2q^{8} + O(q^{10}) \) \( 2q + 2q^{2} - 2q^{5} - 2q^{8} - 2q^{10} - 2q^{11} - 2q^{16} - 2q^{17} - 2q^{22} - 2q^{29} - 2q^{31} - 2q^{34} + 2q^{40} - 2q^{49} + 2q^{55} - 2q^{58} + 2q^{61} - 2q^{62} + 2q^{64} - 2q^{67} + 2q^{80} + 2q^{83} + 2q^{85} + 2q^{88} + 4q^{91} - 2q^{98} + O(q^{100}) \)

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
1.41421i
1.41421i
1.00000 0 0 −1.00000 0 1.41421i −1.00000 0 −1.00000
955.2 1.00000 0 0 −1.00000 0 1.41421i −1.00000 0 −1.00000
\(n\): 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 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2151.1.d.d yes 2
3.b odd 2 1 2151.1.d.b 2
239.b odd 2 1 inner 2151.1.d.d yes 2
717.b even 2 1 2151.1.d.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2151.1.d.b 2 3.b odd 2 1
2151.1.d.b 2 717.b even 2 1
2151.1.d.d yes 2 1.a even 1 1 trivial
2151.1.d.d yes 2 239.b odd 2 1 inner

Hecke kernels

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

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

Hecke characteristic polynomials

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