Properties

Label 87.7.d.a
Level $87$
Weight $7$
Character orbit 87.d
Self dual yes
Analytic conductor $20.015$
Analytic rank $0$
Dimension $1$
CM discriminant -87
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [87,7,Mod(86,87)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(87, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1]))
 
N = Newforms(chi, 7, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("87.86");
 
S:= CuspForms(chi, 7);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 87 = 3 \cdot 29 \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 87.d (of order \(2\), degree \(1\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(20.0147052749\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \( q - 13 q^{2} + 27 q^{3} + 105 q^{4} - 351 q^{6} + 338 q^{7} - 533 q^{8} + 729 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 13 q^{2} + 27 q^{3} + 105 q^{4} - 351 q^{6} + 338 q^{7} - 533 q^{8} + 729 q^{9} + 806 q^{11} + 2835 q^{12} + 3002 q^{13} - 4394 q^{14} + 209 q^{16} - 9778 q^{17} - 9477 q^{18} + 9126 q^{21} - 10478 q^{22} - 14391 q^{24} + 15625 q^{25} - 39026 q^{26} + 19683 q^{27} + 35490 q^{28} - 24389 q^{29} + 31395 q^{32} + 21762 q^{33} + 127114 q^{34} + 76545 q^{36} + 81054 q^{39} + 132158 q^{41} - 118638 q^{42} + 84630 q^{44} + 151502 q^{47} + 5643 q^{48} - 3405 q^{49} - 203125 q^{50} - 264006 q^{51} + 315210 q^{52} - 255879 q^{54} - 180154 q^{56} + 317057 q^{58} + 246402 q^{63} - 421511 q^{64} - 282906 q^{66} + 267098 q^{67} - 1026690 q^{68} - 388557 q^{72} + 421875 q^{75} + 272428 q^{77} - 1053702 q^{78} + 531441 q^{81} - 1718054 q^{82} + 958230 q^{84} - 658503 q^{87} - 429598 q^{88} - 71266 q^{89} + 1014676 q^{91} - 1969526 q^{94} + 847665 q^{96} + 44265 q^{98} + 587574 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
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
−13.0000 27.0000 105.000 0 −351.000 338.000 −533.000 729.000 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 87.7.d.a 1
3.b odd 2 1 87.7.d.b yes 1
29.b even 2 1 87.7.d.b yes 1
87.d odd 2 1 CM 87.7.d.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
87.7.d.a 1 1.a even 1 1 trivial
87.7.d.a 1 87.d odd 2 1 CM
87.7.d.b yes 1 3.b odd 2 1
87.7.d.b yes 1 29.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} + 13 \) acting on \(S_{7}^{\mathrm{new}}(87, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T + 13 \) Copy content Toggle raw display
$3$ \( T - 27 \) Copy content Toggle raw display
$5$ \( T \) Copy content Toggle raw display
$7$ \( T - 338 \) Copy content Toggle raw display
$11$ \( T - 806 \) Copy content Toggle raw display
$13$ \( T - 3002 \) Copy content Toggle raw display
$17$ \( T + 9778 \) Copy content Toggle raw display
$19$ \( T \) Copy content Toggle raw display
$23$ \( T \) Copy content Toggle raw display
$29$ \( T + 24389 \) Copy content Toggle raw display
$31$ \( T \) Copy content Toggle raw display
$37$ \( T \) Copy content Toggle raw display
$41$ \( T - 132158 \) Copy content Toggle raw display
$43$ \( T \) Copy content Toggle raw display
$47$ \( T - 151502 \) Copy content Toggle raw display
$53$ \( T \) Copy content Toggle raw display
$59$ \( T \) Copy content Toggle raw display
$61$ \( T \) Copy content Toggle raw display
$67$ \( T - 267098 \) Copy content Toggle raw display
$71$ \( T \) Copy content Toggle raw display
$73$ \( T \) Copy content Toggle raw display
$79$ \( T \) Copy content Toggle raw display
$83$ \( T \) Copy content Toggle raw display
$89$ \( T + 71266 \) Copy content Toggle raw display
$97$ \( T \) Copy content Toggle raw display
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