Properties

Label 2775.1.b.a
Level $2775$
Weight $1$
Character orbit 2775.b
Analytic conductor $1.385$
Analytic rank $0$
Dimension $2$
Projective image $D_{2}$
CM/RM discs -3, -111, 37
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 2775 = 3 \cdot 5^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2775.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.38490541006\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
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: $D_4:C_2$
Artin field: Galois closure of 8.0.1732640625.1

$q$-expansion

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

\(f(q)\) \(=\) \( q -i q^{3} + q^{4} -2 i q^{7} - q^{9} +O(q^{10})\) \( q -i q^{3} + q^{4} -2 i q^{7} - q^{9} -i q^{12} + q^{16} -2 q^{21} + i q^{27} -2 i q^{28} - q^{36} + i q^{37} -i q^{48} -3 q^{49} + 2 i q^{63} + q^{64} -2 i q^{67} + 2 i q^{73} + q^{81} -2 q^{84} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{4} - 2q^{9} + O(q^{10}) \) \( 2q + 2q^{4} - 2q^{9} + 2q^{16} - 4q^{21} - 2q^{36} - 6q^{49} + 2q^{64} + 2q^{81} - 4q^{84} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2775\mathbb{Z}\right)^\times\).

\(n\) \(76\) \(926\) \(1777\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2774.1
1.00000i
1.00000i
0 1.00000i 1.00000 0 0 2.00000i 0 −1.00000 0
2774.2 0 1.00000i 1.00000 0 0 2.00000i 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 \(\Q(\sqrt{-111}) \)
5.b even 2 1 inner
15.d odd 2 1 inner
185.d even 2 1 inner
555.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2775.1.b.a 2
3.b odd 2 1 CM 2775.1.b.a 2
5.b even 2 1 inner 2775.1.b.a 2
5.c odd 4 1 111.1.d.a 1
5.c odd 4 1 2775.1.h.a 1
15.d odd 2 1 inner 2775.1.b.a 2
15.e even 4 1 111.1.d.a 1
15.e even 4 1 2775.1.h.a 1
20.e even 4 1 1776.1.n.a 1
37.b even 2 1 RM 2775.1.b.a 2
45.k odd 12 2 2997.1.n.b 2
45.l even 12 2 2997.1.n.b 2
60.l odd 4 1 1776.1.n.a 1
111.d odd 2 1 CM 2775.1.b.a 2
185.d even 2 1 inner 2775.1.b.a 2
185.h odd 4 1 111.1.d.a 1
185.h odd 4 1 2775.1.h.a 1
555.b odd 2 1 inner 2775.1.b.a 2
555.n even 4 1 111.1.d.a 1
555.n even 4 1 2775.1.h.a 1
740.m even 4 1 1776.1.n.a 1
1665.dc odd 12 2 2997.1.n.b 2
1665.di even 12 2 2997.1.n.b 2
2220.bf odd 4 1 1776.1.n.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
111.1.d.a 1 5.c odd 4 1
111.1.d.a 1 15.e even 4 1
111.1.d.a 1 185.h odd 4 1
111.1.d.a 1 555.n even 4 1
1776.1.n.a 1 20.e even 4 1
1776.1.n.a 1 60.l odd 4 1
1776.1.n.a 1 740.m even 4 1
1776.1.n.a 1 2220.bf odd 4 1
2775.1.b.a 2 1.a even 1 1 trivial
2775.1.b.a 2 3.b odd 2 1 CM
2775.1.b.a 2 5.b even 2 1 inner
2775.1.b.a 2 15.d odd 2 1 inner
2775.1.b.a 2 37.b even 2 1 RM
2775.1.b.a 2 111.d odd 2 1 CM
2775.1.b.a 2 185.d even 2 1 inner
2775.1.b.a 2 555.b odd 2 1 inner
2775.1.h.a 1 5.c odd 4 1
2775.1.h.a 1 15.e even 4 1
2775.1.h.a 1 185.h odd 4 1
2775.1.h.a 1 555.n even 4 1
2997.1.n.b 2 45.k odd 12 2
2997.1.n.b 2 45.l even 12 2
2997.1.n.b 2 1665.dc odd 12 2
2997.1.n.b 2 1665.di even 12 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} \) acting on \(S_{1}^{\mathrm{new}}(2775, [\chi])\).

Hecke characteristic polynomials

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