Properties

Label 55.1.d.a
Level 55
Weight 1
Character orbit 55.d
Self dual yes
Analytic conductor 0.027
Analytic rank 0
Dimension 1
Projective image \(D_{2}\)
CM/RM discs -11, -55, 5
Inner twists 4

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Newspace parameters

Level: \( N \) \(=\) \( 55 = 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 55.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: yes
Analytic conductor: \(0.0274485756948\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image \(D_{2}\)
Projective field Galois closure of \(\Q(\sqrt{5}, \sqrt{-11})\)
Artin image $D_4$
Artin field Galois closure of 4.2.275.1

$q$-expansion

\(f(q)\) \(=\) \( q - q^{4} - q^{5} + q^{9} + O(q^{10}) \) \( q - q^{4} - q^{5} + q^{9} - q^{11} + q^{16} + q^{20} + q^{25} - 2q^{31} - q^{36} + q^{44} - q^{45} - q^{49} + q^{55} + 2q^{59} - q^{64} + 2q^{71} - q^{80} + q^{81} - 2q^{89} - q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/55\mathbb{Z}\right)^\times\).

\(n\) \(12\) \(46\)
\(\chi(n)\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
54.1
0
0 0 −1.00000 −1.00000 0 0 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
5.b even 2 1 RM by \(\Q(\sqrt{5}) \)
11.b odd 2 1 CM by \(\Q(\sqrt{-11}) \)
55.d odd 2 1 CM by \(\Q(\sqrt{-55}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 55.1.d.a 1
3.b odd 2 1 495.1.h.a 1
4.b odd 2 1 880.1.i.a 1
5.b even 2 1 RM 55.1.d.a 1
5.c odd 4 2 275.1.c.a 1
7.b odd 2 1 2695.1.g.c 1
7.c even 3 2 2695.1.q.c 2
7.d odd 6 2 2695.1.q.b 2
8.b even 2 1 3520.1.i.b 1
8.d odd 2 1 3520.1.i.a 1
11.b odd 2 1 CM 55.1.d.a 1
11.c even 5 4 605.1.h.a 4
11.d odd 10 4 605.1.h.a 4
15.d odd 2 1 495.1.h.a 1
15.e even 4 2 2475.1.b.a 1
20.d odd 2 1 880.1.i.a 1
33.d even 2 1 495.1.h.a 1
35.c odd 2 1 2695.1.g.c 1
35.i odd 6 2 2695.1.q.b 2
35.j even 6 2 2695.1.q.c 2
40.e odd 2 1 3520.1.i.a 1
40.f even 2 1 3520.1.i.b 1
44.c even 2 1 880.1.i.a 1
55.d odd 2 1 CM 55.1.d.a 1
55.e even 4 2 275.1.c.a 1
55.h odd 10 4 605.1.h.a 4
55.j even 10 4 605.1.h.a 4
55.k odd 20 8 3025.1.x.a 4
55.l even 20 8 3025.1.x.a 4
77.b even 2 1 2695.1.g.c 1
77.h odd 6 2 2695.1.q.c 2
77.i even 6 2 2695.1.q.b 2
88.b odd 2 1 3520.1.i.b 1
88.g even 2 1 3520.1.i.a 1
165.d even 2 1 495.1.h.a 1
165.l odd 4 2 2475.1.b.a 1
220.g even 2 1 880.1.i.a 1
385.h even 2 1 2695.1.g.c 1
385.o even 6 2 2695.1.q.b 2
385.q odd 6 2 2695.1.q.c 2
440.c even 2 1 3520.1.i.a 1
440.o odd 2 1 3520.1.i.b 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
55.1.d.a 1 1.a even 1 1 trivial
55.1.d.a 1 5.b even 2 1 RM
55.1.d.a 1 11.b odd 2 1 CM
55.1.d.a 1 55.d odd 2 1 CM
275.1.c.a 1 5.c odd 4 2
275.1.c.a 1 55.e even 4 2
495.1.h.a 1 3.b odd 2 1
495.1.h.a 1 15.d odd 2 1
495.1.h.a 1 33.d even 2 1
495.1.h.a 1 165.d even 2 1
605.1.h.a 4 11.c even 5 4
605.1.h.a 4 11.d odd 10 4
605.1.h.a 4 55.h odd 10 4
605.1.h.a 4 55.j even 10 4
880.1.i.a 1 4.b odd 2 1
880.1.i.a 1 20.d odd 2 1
880.1.i.a 1 44.c even 2 1
880.1.i.a 1 220.g even 2 1
2475.1.b.a 1 15.e even 4 2
2475.1.b.a 1 165.l odd 4 2
2695.1.g.c 1 7.b odd 2 1
2695.1.g.c 1 35.c odd 2 1
2695.1.g.c 1 77.b even 2 1
2695.1.g.c 1 385.h even 2 1
2695.1.q.b 2 7.d odd 6 2
2695.1.q.b 2 35.i odd 6 2
2695.1.q.b 2 77.i even 6 2
2695.1.q.b 2 385.o even 6 2
2695.1.q.c 2 7.c even 3 2
2695.1.q.c 2 35.j even 6 2
2695.1.q.c 2 77.h odd 6 2
2695.1.q.c 2 385.q odd 6 2
3025.1.x.a 4 55.k odd 20 8
3025.1.x.a 4 55.l even 20 8
3520.1.i.a 1 8.d odd 2 1
3520.1.i.a 1 40.e odd 2 1
3520.1.i.a 1 88.g even 2 1
3520.1.i.a 1 440.c even 2 1
3520.1.i.b 1 8.b even 2 1
3520.1.i.b 1 40.f even 2 1
3520.1.i.b 1 88.b odd 2 1
3520.1.i.b 1 440.o odd 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(55, [\chi])\).

Hecke characteristic polynomials

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