Properties

Label 3724.1.e.e
Level $3724$
Weight $1$
Character orbit 3724.e
Self dual yes
Analytic conductor $1.859$
Analytic rank $0$
Dimension $1$
Projective image $D_{3}$
CM discriminant -19
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 3724 = 2^{2} \cdot 7^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3724.e (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: yes
Analytic conductor: \(1.85851810705\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 532)
Projective image: \(D_{3}\)
Projective field: Galois closure of 3.1.3724.1
Artin image: $S_3$
Artin field: Galois closure of 3.1.3724.1

$q$-expansion

\(f(q)\) \(=\) \( q + 2q^{5} + q^{9} + O(q^{10}) \) \( q + 2q^{5} + q^{9} - q^{11} - q^{17} + q^{19} - q^{23} + 3q^{25} + 2q^{43} + 2q^{45} - q^{47} - 2q^{55} - q^{61} - q^{73} + q^{81} - q^{83} - 2q^{85} + 2q^{95} - q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3724\mathbb{Z}\right)^\times\).

\(n\) \(1863\) \(3041\) \(3137\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1177.1
0
0 0 0 2.00000 0 0 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 CM by \(\Q(\sqrt{-19}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3724.1.e.e 1
7.b odd 2 1 3724.1.e.a 1
7.c even 3 2 532.1.bc.a 2
7.d odd 6 2 3724.1.bc.d 2
19.b odd 2 1 CM 3724.1.e.e 1
28.g odd 6 2 2128.1.cl.a 2
133.c even 2 1 3724.1.e.a 1
133.o even 6 2 3724.1.bc.d 2
133.r odd 6 2 532.1.bc.a 2
532.t even 6 2 2128.1.cl.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
532.1.bc.a 2 7.c even 3 2
532.1.bc.a 2 133.r odd 6 2
2128.1.cl.a 2 28.g odd 6 2
2128.1.cl.a 2 532.t even 6 2
3724.1.e.a 1 7.b odd 2 1
3724.1.e.a 1 133.c even 2 1
3724.1.e.e 1 1.a even 1 1 trivial
3724.1.e.e 1 19.b odd 2 1 CM
3724.1.bc.d 2 7.d odd 6 2
3724.1.bc.d 2 133.o even 6 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(3724, [\chi])\):

\( T_{5} - 2 \)
\( T_{11} + 1 \)

Hecke characteristic polynomials

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