Properties

Label 2775.1.x.a
Level $2775$
Weight $1$
Character orbit 2775.x
Analytic conductor $1.385$
Analytic rank $0$
Dimension $4$
Projective image $D_{3}$
CM discriminant -3
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.x (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.38490541006\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \(x^{4} - x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 111)
Projective image: \(D_{3}\)
Projective field: Galois closure of 3.1.4107.1
Artin image: $C_{12}\times S_3$
Artin field: Galois closure of \(\mathbb{Q}[x]/(x^{24} - \cdots)\)

$q$-expansion

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

\(f(q)\) \(=\) \( q + \zeta_{12}^{5} q^{3} + \zeta_{12}^{2} q^{4} + \zeta_{12}^{5} q^{7} -\zeta_{12}^{4} q^{9} +O(q^{10})\) \( q + \zeta_{12}^{5} q^{3} + \zeta_{12}^{2} q^{4} + \zeta_{12}^{5} q^{7} -\zeta_{12}^{4} q^{9} -\zeta_{12} q^{12} -\zeta_{12}^{5} q^{13} + \zeta_{12}^{4} q^{16} + 2 \zeta_{12}^{2} q^{19} -\zeta_{12}^{4} q^{21} + \zeta_{12}^{3} q^{27} -\zeta_{12} q^{28} - q^{31} + q^{36} + \zeta_{12}^{3} q^{37} + \zeta_{12}^{4} q^{39} + \zeta_{12}^{3} q^{43} -\zeta_{12}^{3} q^{48} + \zeta_{12} q^{52} -2 \zeta_{12} q^{57} -2 \zeta_{12}^{2} q^{61} + \zeta_{12}^{3} q^{63} - q^{64} + \zeta_{12}^{5} q^{67} + \zeta_{12}^{3} q^{73} + 2 \zeta_{12}^{4} q^{76} -\zeta_{12}^{2} q^{79} -\zeta_{12}^{2} q^{81} + q^{84} + \zeta_{12}^{4} q^{91} -\zeta_{12}^{5} q^{93} -\zeta_{12}^{3} q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{4} + 2 q^{9} + O(q^{10}) \) \( 4 q + 2 q^{4} + 2 q^{9} - 2 q^{16} + 4 q^{19} + 2 q^{21} - 4 q^{31} + 4 q^{36} - 2 q^{39} - 4 q^{61} - 4 q^{64} - 4 q^{76} - 2 q^{79} - 2 q^{81} + 4 q^{84} - 2 q^{91} + O(q^{100}) \)

Character values

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

\(n\) \(76\) \(926\) \(1777\)
\(\chi(n)\) \(-\zeta_{12}^{2}\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
824.1
0.866025 + 0.500000i
−0.866025 0.500000i
0.866025 0.500000i
−0.866025 + 0.500000i
0 −0.866025 + 0.500000i 0.500000 + 0.866025i 0 0 −0.866025 + 0.500000i 0 0.500000 0.866025i 0
824.2 0 0.866025 0.500000i 0.500000 + 0.866025i 0 0 0.866025 0.500000i 0 0.500000 0.866025i 0
2024.1 0 −0.866025 0.500000i 0.500000 0.866025i 0 0 −0.866025 0.500000i 0 0.500000 + 0.866025i 0
2024.2 0 0.866025 + 0.500000i 0.500000 0.866025i 0 0 0.866025 + 0.500000i 0 0.500000 + 0.866025i 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 \(\Q(\sqrt{-3}) \)
5.b even 2 1 inner
15.d odd 2 1 inner
37.c even 3 1 inner
111.i odd 6 1 inner
185.n even 6 1 inner
555.w odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2775.1.x.a 4
3.b odd 2 1 CM 2775.1.x.a 4
5.b even 2 1 inner 2775.1.x.a 4
5.c odd 4 1 111.1.i.a 2
5.c odd 4 1 2775.1.z.a 2
15.d odd 2 1 inner 2775.1.x.a 4
15.e even 4 1 111.1.i.a 2
15.e even 4 1 2775.1.z.a 2
20.e even 4 1 1776.1.bw.a 2
37.c even 3 1 inner 2775.1.x.a 4
45.k odd 12 1 2997.1.l.a 2
45.k odd 12 1 2997.1.u.a 2
45.l even 12 1 2997.1.l.a 2
45.l even 12 1 2997.1.u.a 2
60.l odd 4 1 1776.1.bw.a 2
111.i odd 6 1 inner 2775.1.x.a 4
185.n even 6 1 inner 2775.1.x.a 4
185.s odd 12 1 111.1.i.a 2
185.s odd 12 1 2775.1.z.a 2
555.w odd 6 1 inner 2775.1.x.a 4
555.bi even 12 1 111.1.i.a 2
555.bi even 12 1 2775.1.z.a 2
740.bg even 12 1 1776.1.bw.a 2
1665.co odd 12 1 2997.1.u.a 2
1665.cp even 12 1 2997.1.u.a 2
1665.de odd 12 1 2997.1.l.a 2
1665.dg even 12 1 2997.1.l.a 2
2220.cu odd 12 1 1776.1.bw.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
111.1.i.a 2 5.c odd 4 1
111.1.i.a 2 15.e even 4 1
111.1.i.a 2 185.s odd 12 1
111.1.i.a 2 555.bi even 12 1
1776.1.bw.a 2 20.e even 4 1
1776.1.bw.a 2 60.l odd 4 1
1776.1.bw.a 2 740.bg even 12 1
1776.1.bw.a 2 2220.cu odd 12 1
2775.1.x.a 4 1.a even 1 1 trivial
2775.1.x.a 4 3.b odd 2 1 CM
2775.1.x.a 4 5.b even 2 1 inner
2775.1.x.a 4 15.d odd 2 1 inner
2775.1.x.a 4 37.c even 3 1 inner
2775.1.x.a 4 111.i odd 6 1 inner
2775.1.x.a 4 185.n even 6 1 inner
2775.1.x.a 4 555.w odd 6 1 inner
2775.1.z.a 2 5.c odd 4 1
2775.1.z.a 2 15.e even 4 1
2775.1.z.a 2 185.s odd 12 1
2775.1.z.a 2 555.bi even 12 1
2997.1.l.a 2 45.k odd 12 1
2997.1.l.a 2 45.l even 12 1
2997.1.l.a 2 1665.de odd 12 1
2997.1.l.a 2 1665.dg even 12 1
2997.1.u.a 2 45.k odd 12 1
2997.1.u.a 2 45.l even 12 1
2997.1.u.a 2 1665.co odd 12 1
2997.1.u.a 2 1665.cp even 12 1

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(2775, [\chi])\).

Hecke characteristic polynomials

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