Properties

Label 1444.1.c.a
Level $1444$
Weight $1$
Character orbit 1444.c
Analytic conductor $0.721$
Analytic rank $0$
Dimension $2$
Projective image $S_{4}$
CM/RM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1444 = 2^{2} \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1444.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.720649878242\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-2}) \)
Defining polynomial: \(x^{2} + 2\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(S_{4}\)
Projective field: Galois closure of 4.2.27436.1
Artin image: $\GL(2,3)$
Artin field: Galois closure of 8.2.14301947824.1

$q$-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.

\(f(q)\) \(=\) \( q -\beta q^{3} + q^{5} + q^{7} - q^{9} +O(q^{10})\) \( q -\beta q^{3} + q^{5} + q^{7} - q^{9} - q^{11} -\beta q^{15} + q^{17} -\beta q^{21} + \beta q^{29} + \beta q^{33} + q^{35} -\beta q^{37} + \beta q^{41} - q^{43} - q^{45} - q^{47} -\beta q^{51} - q^{55} + \beta q^{59} - q^{61} - q^{63} -\beta q^{71} + q^{73} - q^{77} - q^{81} + q^{85} + 2 q^{87} + q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{5} + 2q^{7} - 2q^{9} + O(q^{10}) \) \( 2q + 2q^{5} + 2q^{7} - 2q^{9} - 2q^{11} + 2q^{17} + 2q^{35} - 2q^{43} - 2q^{45} - 2q^{47} - 2q^{55} - 2q^{61} - 2q^{63} + 2q^{73} - 2q^{77} - 2q^{81} + 2q^{85} + 4q^{87} + 2q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1444\mathbb{Z}\right)^\times\).

\(n\) \(723\) \(1085\)
\(\chi(n)\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
721.1
1.41421i
1.41421i
0 1.41421i 0 1.00000 0 1.00000 0 −1.00000 0
721.2 0 1.41421i 0 1.00000 0 1.00000 0 −1.00000 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
19.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1444.1.c.a 2
19.b odd 2 1 inner 1444.1.c.a 2
19.c even 3 2 1444.1.h.b 4
19.d odd 6 2 1444.1.h.b 4
19.e even 9 6 1444.1.j.b 12
19.f odd 18 6 1444.1.j.b 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1444.1.c.a 2 1.a even 1 1 trivial
1444.1.c.a 2 19.b odd 2 1 inner
1444.1.h.b 4 19.c even 3 2
1444.1.h.b 4 19.d odd 6 2
1444.1.j.b 12 19.e even 9 6
1444.1.j.b 12 19.f odd 18 6

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(1444, [\chi])\).

Hecke characteristic polynomials

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