Properties

Label 87.1.d.b
Level 87
Weight 1
Character orbit 87.d
Self dual yes
Analytic conductor 0.043
Analytic rank 0
Dimension 1
Projective image \(D_{3}\)
CM discriminant -87
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 87 = 3 \cdot 29 \)
Weight: \( k \) = \( 1 \)
Character orbit: \([\chi]\) = 87.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: yes
Analytic conductor: \(0.0434186560991\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image \(D_{3}\)
Projective field Galois closure of 3.1.87.1
Artin image $D_6$
Artin field Galois closure of 6.0.22707.1

$q$-expansion

\(f(q)\) \(=\) \( q + q^{2} - q^{3} - q^{6} - q^{7} - q^{8} + q^{9} + O(q^{10}) \) \( q + q^{2} - q^{3} - q^{6} - q^{7} - q^{8} + q^{9} + q^{11} - q^{13} - q^{14} - q^{16} + q^{17} + q^{18} + q^{21} + q^{22} + q^{24} + q^{25} - q^{26} - q^{27} - q^{29} - q^{33} + q^{34} + q^{39} - 2q^{41} + q^{42} + q^{47} + q^{48} + q^{50} - q^{51} - q^{54} + q^{56} - q^{58} - q^{63} + q^{64} - q^{66} - q^{67} - q^{72} - q^{75} - q^{77} + q^{78} + q^{81} - 2q^{82} + q^{87} - q^{88} + q^{89} + q^{91} + q^{94} + q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/87\mathbb{Z}\right)^\times\).

\(n\) \(31\) \(59\)
\(\chi(n)\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
86.1
0
1.00000 −1.00000 0 0 −1.00000 −1.00000 −1.00000 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
87.d odd 2 1 CM by \(\Q(\sqrt{-87}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 87.1.d.b yes 1
3.b odd 2 1 87.1.d.a 1
4.b odd 2 1 1392.1.i.b 1
5.b even 2 1 2175.1.h.a 1
5.c odd 4 2 2175.1.b.b 2
9.c even 3 2 2349.1.h.a 2
9.d odd 6 2 2349.1.h.b 2
12.b even 2 1 1392.1.i.a 1
15.d odd 2 1 2175.1.h.b 1
15.e even 4 2 2175.1.b.a 2
29.b even 2 1 87.1.d.a 1
29.c odd 4 2 2523.1.b.b 2
29.d even 7 6 2523.1.h.a 6
29.e even 14 6 2523.1.h.b 6
29.f odd 28 12 2523.1.j.b 12
87.d odd 2 1 CM 87.1.d.b yes 1
87.f even 4 2 2523.1.b.b 2
87.h odd 14 6 2523.1.h.a 6
87.j odd 14 6 2523.1.h.b 6
87.k even 28 12 2523.1.j.b 12
116.d odd 2 1 1392.1.i.a 1
145.d even 2 1 2175.1.h.b 1
145.h odd 4 2 2175.1.b.a 2
261.h odd 6 2 2349.1.h.a 2
261.i even 6 2 2349.1.h.b 2
348.b even 2 1 1392.1.i.b 1
435.b odd 2 1 2175.1.h.a 1
435.p even 4 2 2175.1.b.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
87.1.d.a 1 3.b odd 2 1
87.1.d.a 1 29.b even 2 1
87.1.d.b yes 1 1.a even 1 1 trivial
87.1.d.b yes 1 87.d odd 2 1 CM
1392.1.i.a 1 12.b even 2 1
1392.1.i.a 1 116.d odd 2 1
1392.1.i.b 1 4.b odd 2 1
1392.1.i.b 1 348.b even 2 1
2175.1.b.a 2 15.e even 4 2
2175.1.b.a 2 145.h odd 4 2
2175.1.b.b 2 5.c odd 4 2
2175.1.b.b 2 435.p even 4 2
2175.1.h.a 1 5.b even 2 1
2175.1.h.a 1 435.b odd 2 1
2175.1.h.b 1 15.d odd 2 1
2175.1.h.b 1 145.d even 2 1
2349.1.h.a 2 9.c even 3 2
2349.1.h.a 2 261.h odd 6 2
2349.1.h.b 2 9.d odd 6 2
2349.1.h.b 2 261.i even 6 2
2523.1.b.b 2 29.c odd 4 2
2523.1.b.b 2 87.f even 4 2
2523.1.h.a 6 29.d even 7 6
2523.1.h.a 6 87.h odd 14 6
2523.1.h.b 6 29.e even 14 6
2523.1.h.b 6 87.j odd 14 6
2523.1.j.b 12 29.f odd 28 12
2523.1.j.b 12 87.k even 28 12

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} - 1 \) acting on \(S_{1}^{\mathrm{new}}(87, [\chi])\).

Hecke Characteristic Polynomials

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