Properties

Label 1872.1.bd.a
Level $1872$
Weight $1$
Character orbit 1872.bd
Analytic conductor $0.934$
Analytic rank $0$
Dimension $2$
Projective image $D_{4}$
CM discriminant -3
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 1872 = 2^{4} \cdot 3^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1872.bd (of order \(4\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.934249703649\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 117)
Projective image: \(D_{4}\)
Projective field: Galois closure of 4.2.6591.1
Artin image: $C_2\times C_4{\rm wrC}_2$
Artin field: Galois closure of \(\mathbb{Q}[x]/(x^{16} + \cdots)\)

$q$-expansion

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

\(f(q)\) \(=\) \( q + ( 1 - i ) q^{7} +O(q^{10})\) \( q + ( 1 - i ) q^{7} + i q^{13} + ( 1 + i ) q^{19} -i q^{25} + ( -1 - i ) q^{31} + ( 1 - i ) q^{37} -i q^{49} + ( 1 + i ) q^{67} + ( -1 + i ) q^{73} + ( 1 + i ) q^{91} + ( 1 + i ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{7} + O(q^{10}) \) \( 2q + 2q^{7} + 2q^{19} - 2q^{31} + 2q^{37} + 2q^{67} - 2q^{73} + 2q^{91} + 2q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1872\mathbb{Z}\right)^\times\).

\(n\) \(145\) \(209\) \(469\) \(703\)
\(\chi(n)\) \(i\) \(1\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
577.1
1.00000i
1.00000i
0 0 0 0 0 1.00000 + 1.00000i 0 0 0
1009.1 0 0 0 0 0 1.00000 1.00000i 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 CM by \(\Q(\sqrt{-3}) \)
13.d odd 4 1 inner
39.f even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1872.1.bd.a 2
3.b odd 2 1 CM 1872.1.bd.a 2
4.b odd 2 1 117.1.j.a 2
12.b even 2 1 117.1.j.a 2
13.d odd 4 1 inner 1872.1.bd.a 2
20.d odd 2 1 2925.1.s.a 2
20.e even 4 1 2925.1.t.a 2
20.e even 4 1 2925.1.t.b 2
36.f odd 6 2 1053.1.bb.a 4
36.h even 6 2 1053.1.bb.a 4
39.f even 4 1 inner 1872.1.bd.a 2
52.b odd 2 1 1521.1.j.b 2
52.f even 4 1 117.1.j.a 2
52.f even 4 1 1521.1.j.b 2
52.i odd 6 2 1521.1.bd.b 4
52.j odd 6 2 1521.1.bd.c 4
52.l even 12 2 1521.1.bd.b 4
52.l even 12 2 1521.1.bd.c 4
60.h even 2 1 2925.1.s.a 2
60.l odd 4 1 2925.1.t.a 2
60.l odd 4 1 2925.1.t.b 2
156.h even 2 1 1521.1.j.b 2
156.l odd 4 1 117.1.j.a 2
156.l odd 4 1 1521.1.j.b 2
156.p even 6 2 1521.1.bd.c 4
156.r even 6 2 1521.1.bd.b 4
156.v odd 12 2 1521.1.bd.b 4
156.v odd 12 2 1521.1.bd.c 4
260.l odd 4 1 2925.1.t.a 2
260.s odd 4 1 2925.1.t.b 2
260.u even 4 1 2925.1.s.a 2
468.bs even 12 2 1053.1.bb.a 4
468.ch odd 12 2 1053.1.bb.a 4
780.u even 4 1 2925.1.t.a 2
780.bb odd 4 1 2925.1.s.a 2
780.bn even 4 1 2925.1.t.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
117.1.j.a 2 4.b odd 2 1
117.1.j.a 2 12.b even 2 1
117.1.j.a 2 52.f even 4 1
117.1.j.a 2 156.l odd 4 1
1053.1.bb.a 4 36.f odd 6 2
1053.1.bb.a 4 36.h even 6 2
1053.1.bb.a 4 468.bs even 12 2
1053.1.bb.a 4 468.ch odd 12 2
1521.1.j.b 2 52.b odd 2 1
1521.1.j.b 2 52.f even 4 1
1521.1.j.b 2 156.h even 2 1
1521.1.j.b 2 156.l odd 4 1
1521.1.bd.b 4 52.i odd 6 2
1521.1.bd.b 4 52.l even 12 2
1521.1.bd.b 4 156.r even 6 2
1521.1.bd.b 4 156.v odd 12 2
1521.1.bd.c 4 52.j odd 6 2
1521.1.bd.c 4 52.l even 12 2
1521.1.bd.c 4 156.p even 6 2
1521.1.bd.c 4 156.v odd 12 2
1872.1.bd.a 2 1.a even 1 1 trivial
1872.1.bd.a 2 3.b odd 2 1 CM
1872.1.bd.a 2 13.d odd 4 1 inner
1872.1.bd.a 2 39.f even 4 1 inner
2925.1.s.a 2 20.d odd 2 1
2925.1.s.a 2 60.h even 2 1
2925.1.s.a 2 260.u even 4 1
2925.1.s.a 2 780.bb odd 4 1
2925.1.t.a 2 20.e even 4 1
2925.1.t.a 2 60.l odd 4 1
2925.1.t.a 2 260.l odd 4 1
2925.1.t.a 2 780.u even 4 1
2925.1.t.b 2 20.e even 4 1
2925.1.t.b 2 60.l odd 4 1
2925.1.t.b 2 260.s odd 4 1
2925.1.t.b 2 780.bn even 4 1

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(1872, [\chi])\).

Hecke characteristic polynomials

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