Properties

Label 3744.1.bd.b
Level $3744$
Weight $1$
Character orbit 3744.bd
Analytic conductor $1.868$
Analytic rank $0$
Dimension $2$
Projective image $D_{4}$
CM discriminant -4
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 3744 = 2^{5} \cdot 3^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3744.bd (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.86849940730\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 416)
Projective image \(D_{4}\)
Projective field Galois closure of 4.2.35152.1
Artin image $C_2\times C_4\wr C_2$
Artin field Galois closure of \(\mathbb{Q}[x]/(x^{16} - \cdots)\)

$q$-expansion

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

\(f(q)\) \(=\) \( q + ( -1 - i ) q^{5} +O(q^{10})\) \( q + ( -1 - i ) q^{5} + i q^{13} + 2 i q^{17} + i q^{25} + ( 1 - i ) q^{37} + ( 1 + i ) q^{41} + i q^{49} + 2 q^{53} -2 q^{61} + ( 1 - i ) q^{65} + ( 1 - i ) q^{73} + ( 2 - 2 i ) q^{85} + ( 1 - i ) q^{89} + ( 1 + i ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{5} + O(q^{10}) \) \( 2q - 2q^{5} + 2q^{37} + 2q^{41} + 4q^{53} - 4q^{61} + 2q^{65} + 2q^{73} + 4q^{85} + 2q^{89} + 2q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3744\mathbb{Z}\right)^\times\).

\(n\) \(703\) \(2017\) \(2081\) \(2341\)
\(\chi(n)\) \(1\) \(i\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
577.1
1.00000i
1.00000i
0 0 0 −1.00000 + 1.00000i 0 0 0 0 0
2881.1 0 0 0 −1.00000 1.00000i 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
4.b odd 2 1 CM by \(\Q(\sqrt{-1}) \)
13.d odd 4 1 inner
52.f even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3744.1.bd.b 2
3.b odd 2 1 416.1.t.a 2
4.b odd 2 1 CM 3744.1.bd.b 2
12.b even 2 1 416.1.t.a 2
13.d odd 4 1 inner 3744.1.bd.b 2
24.f even 2 1 832.1.t.a 2
24.h odd 2 1 832.1.t.a 2
39.f even 4 1 416.1.t.a 2
48.i odd 4 1 3328.1.j.a 2
48.i odd 4 1 3328.1.j.b 2
48.k even 4 1 3328.1.j.a 2
48.k even 4 1 3328.1.j.b 2
52.f even 4 1 inner 3744.1.bd.b 2
156.l odd 4 1 416.1.t.a 2
312.w odd 4 1 832.1.t.a 2
312.y even 4 1 832.1.t.a 2
624.s odd 4 1 3328.1.j.b 2
624.u even 4 1 3328.1.j.b 2
624.bm even 4 1 3328.1.j.a 2
624.bo odd 4 1 3328.1.j.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
416.1.t.a 2 3.b odd 2 1
416.1.t.a 2 12.b even 2 1
416.1.t.a 2 39.f even 4 1
416.1.t.a 2 156.l odd 4 1
832.1.t.a 2 24.f even 2 1
832.1.t.a 2 24.h odd 2 1
832.1.t.a 2 312.w odd 4 1
832.1.t.a 2 312.y even 4 1
3328.1.j.a 2 48.i odd 4 1
3328.1.j.a 2 48.k even 4 1
3328.1.j.a 2 624.bm even 4 1
3328.1.j.a 2 624.bo odd 4 1
3328.1.j.b 2 48.i odd 4 1
3328.1.j.b 2 48.k even 4 1
3328.1.j.b 2 624.s odd 4 1
3328.1.j.b 2 624.u even 4 1
3744.1.bd.b 2 1.a even 1 1 trivial
3744.1.bd.b 2 4.b odd 2 1 CM
3744.1.bd.b 2 13.d odd 4 1 inner
3744.1.bd.b 2 52.f even 4 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(3744, [\chi])\):

\( T_{5}^{2} + 2 T_{5} + 2 \)
\( T_{11} \)
\( T_{17}^{2} + 4 \)

Hecke characteristic polynomials

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