Properties

Label 3724.1.bc.d
Level $3724$
Weight $1$
Character orbit 3724.bc
Analytic conductor $1.859$
Analytic rank $0$
Dimension $2$
Projective image $D_{3}$
CM discriminant -19
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 3724 = 2^{2} \cdot 7^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3724.bc (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.85851810705\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
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: $C_6\times S_3$
Artin field: Galois closure of \(\mathbb{Q}[x]/(x^{12} - \cdots)\)

$q$-expansion

The \(q\)-expansion and trace form are shown below.

\(f(q)\) \(=\) \( q + 2 \zeta_{6} q^{5} -\zeta_{6} q^{9} +O(q^{10})\) \( q + 2 \zeta_{6} q^{5} -\zeta_{6} q^{9} -\zeta_{6}^{2} q^{11} + \zeta_{6}^{2} q^{17} + \zeta_{6} q^{19} + \zeta_{6} q^{23} + 3 \zeta_{6}^{2} q^{25} + 2 q^{43} -2 \zeta_{6}^{2} q^{45} -\zeta_{6} q^{47} + 2 q^{55} -\zeta_{6} q^{61} + \zeta_{6}^{2} q^{73} + \zeta_{6}^{2} q^{81} + q^{83} -2 q^{85} + 2 \zeta_{6}^{2} q^{95} - q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{5} - q^{9} + O(q^{10}) \) \( 2q + 2q^{5} - q^{9} + q^{11} - q^{17} + q^{19} + q^{23} - 3q^{25} + 4q^{43} + 2q^{45} - q^{47} + 4q^{55} - q^{61} - q^{73} - q^{81} + 2q^{83} - 4q^{85} - 2q^{95} - 2q^{99} + O(q^{100}) \)

Character values

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

\(n\) \(1863\) \(3041\) \(3137\)
\(\chi(n)\) \(1\) \(-\zeta_{6}\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
569.1
0.500000 0.866025i
0.500000 + 0.866025i
0 0 0 1.00000 1.73205i 0 0 0 −0.500000 + 0.866025i 0
949.1 0 0 0 1.00000 + 1.73205i 0 0 0 −0.500000 0.866025i 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 \(\Q(\sqrt{-19}) \)
7.c even 3 1 inner
133.r odd 6 1 inner

Twists

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

Hecke kernels

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

\( T_{5}^{2} - 2 T_{5} + 4 \)
\( T_{11}^{2} - T_{11} + 1 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( T^{2} \)
$5$ \( 4 - 2 T + T^{2} \)
$7$ \( T^{2} \)
$11$ \( 1 - T + T^{2} \)
$13$ \( T^{2} \)
$17$ \( 1 + T + T^{2} \)
$19$ \( 1 - T + T^{2} \)
$23$ \( 1 - T + T^{2} \)
$29$ \( T^{2} \)
$31$ \( T^{2} \)
$37$ \( T^{2} \)
$41$ \( T^{2} \)
$43$ \( ( -2 + T )^{2} \)
$47$ \( 1 + T + T^{2} \)
$53$ \( T^{2} \)
$59$ \( T^{2} \)
$61$ \( 1 + T + T^{2} \)
$67$ \( T^{2} \)
$71$ \( T^{2} \)
$73$ \( 1 + T + T^{2} \)
$79$ \( T^{2} \)
$83$ \( ( -1 + T )^{2} \)
$89$ \( T^{2} \)
$97$ \( T^{2} \)
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