Properties

Label 3776.1.h.b
Level $3776$
Weight $1$
Character orbit 3776.h
Self dual yes
Analytic conductor $1.884$
Analytic rank $0$
Dimension $1$
Projective image $D_{3}$
CM discriminant -59
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 3776 = 2^{6} \cdot 59 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3776.h (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: yes
Analytic conductor: \(1.88446948770\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 59)
Projective image \(D_{3}\)
Projective field Galois closure of 3.1.59.1
Artin image $D_6$
Artin field Galois closure of 6.2.1782272.1

$q$-expansion

\(f(q)\) \(=\) \( q + q^{3} + q^{5} - q^{7} + O(q^{10}) \) \( q + q^{3} + q^{5} - q^{7} + q^{15} + 2q^{17} + q^{19} - q^{21} - q^{27} + q^{29} - q^{35} - q^{41} + 2q^{51} + q^{53} + q^{57} - q^{59} + 2q^{71} - q^{79} - q^{81} + 2q^{85} + q^{87} + q^{95} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3776\mathbb{Z}\right)^\times\).

\(n\) \(709\) \(769\) \(1535\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2241.1
0
0 1.00000 0 1.00000 0 −1.00000 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
59.b odd 2 1 CM by \(\Q(\sqrt{-59}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3776.1.h.b 1
4.b odd 2 1 3776.1.h.a 1
8.b even 2 1 59.1.b.a 1
8.d odd 2 1 944.1.h.a 1
24.h odd 2 1 531.1.c.a 1
40.f even 2 1 1475.1.c.b 1
40.i odd 4 2 1475.1.d.a 2
56.h odd 2 1 2891.1.c.e 1
56.j odd 6 2 2891.1.g.b 2
56.p even 6 2 2891.1.g.d 2
59.b odd 2 1 CM 3776.1.h.b 1
236.c even 2 1 3776.1.h.a 1
472.c odd 2 1 59.1.b.a 1
472.f even 2 1 944.1.h.a 1
472.o odd 58 28 3481.1.d.a 28
472.p even 58 28 3481.1.d.a 28
1416.o even 2 1 531.1.c.a 1
2360.l odd 2 1 1475.1.c.b 1
2360.w even 4 2 1475.1.d.a 2
3304.p even 2 1 2891.1.c.e 1
3304.r even 6 2 2891.1.g.b 2
3304.bd odd 6 2 2891.1.g.d 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
59.1.b.a 1 8.b even 2 1
59.1.b.a 1 472.c odd 2 1
531.1.c.a 1 24.h odd 2 1
531.1.c.a 1 1416.o even 2 1
944.1.h.a 1 8.d odd 2 1
944.1.h.a 1 472.f even 2 1
1475.1.c.b 1 40.f even 2 1
1475.1.c.b 1 2360.l odd 2 1
1475.1.d.a 2 40.i odd 4 2
1475.1.d.a 2 2360.w even 4 2
2891.1.c.e 1 56.h odd 2 1
2891.1.c.e 1 3304.p even 2 1
2891.1.g.b 2 56.j odd 6 2
2891.1.g.b 2 3304.r even 6 2
2891.1.g.d 2 56.p even 6 2
2891.1.g.d 2 3304.bd odd 6 2
3481.1.d.a 28 472.o odd 58 28
3481.1.d.a 28 472.p even 58 28
3776.1.h.a 1 4.b odd 2 1
3776.1.h.a 1 236.c even 2 1
3776.1.h.b 1 1.a even 1 1 trivial
3776.1.h.b 1 59.b odd 2 1 CM

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(3776, [\chi])\):

\( T_{3} - 1 \)
\( T_{7} + 1 \)

Hecke characteristic polynomials

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