Properties

Label 1776.1.n.a
Level $1776$
Weight $1$
Character orbit 1776.n
Self dual yes
Analytic conductor $0.886$
Analytic rank $0$
Dimension $1$
Projective image $D_{2}$
CM/RM discs -3, -111, 37
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 1776 = 2^{4} \cdot 3 \cdot 37 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1776.n (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: yes
Analytic conductor: \(0.886339462436\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 111)
Projective image: \(D_{2}\)
Projective field: Galois closure of \(\Q(\sqrt{-3}, \sqrt{37})\)
Artin image: $D_4$
Artin field: Galois closure of 4.0.5328.1

$q$-expansion

\(f(q)\) \(=\) \( q - q^{3} + 2q^{7} + q^{9} + O(q^{10}) \) \( q - q^{3} + 2q^{7} + q^{9} - 2q^{21} - q^{25} - q^{27} + q^{37} + 3q^{49} + 2q^{63} + 2q^{67} - 2q^{73} + q^{75} + q^{81} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1776\mathbb{Z}\right)^\times\).

\(n\) \(223\) \(593\) \(1297\) \(1333\)
\(\chi(n)\) \(1\) \(-1\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1553.1
0
0 −1.00000 0 0 0 2.00000 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
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)
37.b even 2 1 RM by \(\Q(\sqrt{37}) \)
111.d odd 2 1 CM by \(\Q(\sqrt{-111}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1776.1.n.a 1
3.b odd 2 1 CM 1776.1.n.a 1
4.b odd 2 1 111.1.d.a 1
12.b even 2 1 111.1.d.a 1
20.d odd 2 1 2775.1.h.a 1
20.e even 4 2 2775.1.b.a 2
36.f odd 6 2 2997.1.n.b 2
36.h even 6 2 2997.1.n.b 2
37.b even 2 1 RM 1776.1.n.a 1
60.h even 2 1 2775.1.h.a 1
60.l odd 4 2 2775.1.b.a 2
111.d odd 2 1 CM 1776.1.n.a 1
148.b odd 2 1 111.1.d.a 1
444.g even 2 1 111.1.d.a 1
740.g odd 2 1 2775.1.h.a 1
740.m even 4 2 2775.1.b.a 2
1332.z even 6 2 2997.1.n.b 2
1332.bl odd 6 2 2997.1.n.b 2
2220.p even 2 1 2775.1.h.a 1
2220.bf odd 4 2 2775.1.b.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
111.1.d.a 1 4.b odd 2 1
111.1.d.a 1 12.b even 2 1
111.1.d.a 1 148.b odd 2 1
111.1.d.a 1 444.g even 2 1
1776.1.n.a 1 1.a even 1 1 trivial
1776.1.n.a 1 3.b odd 2 1 CM
1776.1.n.a 1 37.b even 2 1 RM
1776.1.n.a 1 111.d odd 2 1 CM
2775.1.b.a 2 20.e even 4 2
2775.1.b.a 2 60.l odd 4 2
2775.1.b.a 2 740.m even 4 2
2775.1.b.a 2 2220.bf odd 4 2
2775.1.h.a 1 20.d odd 2 1
2775.1.h.a 1 60.h even 2 1
2775.1.h.a 1 740.g odd 2 1
2775.1.h.a 1 2220.p even 2 1
2997.1.n.b 2 36.f odd 6 2
2997.1.n.b 2 36.h even 6 2
2997.1.n.b 2 1332.z even 6 2
2997.1.n.b 2 1332.bl odd 6 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(1776, [\chi])\):

\( T_{5} \)
\( T_{7} - 2 \)

Hecke characteristic polynomials

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