Properties

Label 1944.1.e.a
Level $1944$
Weight $1$
Character orbit 1944.e
Analytic conductor $0.970$
Analytic rank $0$
Dimension $2$
Projective image $S_{4}$
CM/RM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1944 = 2^{3} \cdot 3^{5} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1944.e (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.970182384559\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-2}) \)
Defining polynomial: \(x^{2} + 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(S_{4}\)
Projective field: Galois closure of 4.2.3888.1
Artin image: $\GL(2,3)$
Artin field: Galois closure of 8.2.181398528.1

$q$-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.

\(f(q)\) \(=\) \( q -\beta q^{5} +O(q^{10})\) \( q -\beta q^{5} -\beta q^{11} + q^{13} + \beta q^{17} - q^{19} -\beta q^{23} - q^{25} - q^{31} + q^{43} - q^{49} -\beta q^{53} -2 q^{55} -\beta q^{59} + q^{61} -\beta q^{65} - q^{67} + \beta q^{71} + q^{73} + q^{79} + \beta q^{83} + 2 q^{85} + \beta q^{95} - q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + O(q^{10}) \) \( 2 q + 2 q^{13} - 2 q^{19} - 2 q^{25} - 2 q^{31} + 2 q^{43} - 2 q^{49} - 4 q^{55} + 2 q^{61} - 2 q^{67} + 2 q^{73} + 2 q^{79} + 4 q^{85} - 2 q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1944\mathbb{Z}\right)^\times\).

\(n\) \(487\) \(973\) \(1217\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1457.1
1.41421i
1.41421i
0 0 0 1.41421i 0 0 0 0 0
1457.2 0 0 0 1.41421i 0 0 0 0 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
3.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1944.1.e.a 2
3.b odd 2 1 inner 1944.1.e.a 2
4.b odd 2 1 3888.1.e.d 2
9.c even 3 2 1944.1.m.a 4
9.d odd 6 2 1944.1.m.a 4
12.b even 2 1 3888.1.e.d 2
36.f odd 6 2 3888.1.q.d 4
36.h even 6 2 3888.1.q.d 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1944.1.e.a 2 1.a even 1 1 trivial
1944.1.e.a 2 3.b odd 2 1 inner
1944.1.m.a 4 9.c even 3 2
1944.1.m.a 4 9.d odd 6 2
3888.1.e.d 2 4.b odd 2 1
3888.1.e.d 2 12.b even 2 1
3888.1.q.d 4 36.f odd 6 2
3888.1.q.d 4 36.h even 6 2

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(1944, [\chi])\).

Hecke characteristic polynomials

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