Properties

Label 15.9.d.a
Level $15$
Weight $9$
Character orbit 15.d
Self dual yes
Analytic conductor $6.111$
Analytic rank $0$
Dimension $1$
CM discriminant -15
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [15,9,Mod(14,15)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(15, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1]))
 
N = Newforms(chi, 9, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("15.14");
 
S:= CuspForms(chi, 9);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 15 = 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 15.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: \(6.11067915092\)
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)
 
\(f(q)\) \(=\) \( q - 17 q^{2} - 81 q^{3} + 33 q^{4} - 625 q^{5} + 1377 q^{6} + 3791 q^{8} + 6561 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 17 q^{2} - 81 q^{3} + 33 q^{4} - 625 q^{5} + 1377 q^{6} + 3791 q^{8} + 6561 q^{9} + 10625 q^{10} - 2673 q^{12} + 50625 q^{15} - 72895 q^{16} + 21118 q^{17} - 111537 q^{18} - 203998 q^{19} - 20625 q^{20} + 550078 q^{23} - 307071 q^{24} + 390625 q^{25} - 531441 q^{27} - 860625 q^{30} + 1831682 q^{31} + 268719 q^{32} - 359006 q^{34} + 216513 q^{36} + 3467966 q^{38} - 2369375 q^{40} - 4100625 q^{45} - 9351326 q^{46} - 8065922 q^{47} + 5904495 q^{48} + 5764801 q^{49} - 6640625 q^{50} - 1710558 q^{51} + 12619678 q^{53} + 9034497 q^{54} + 16523838 q^{57} + 1670625 q^{60} + 14324642 q^{61} - 31138594 q^{62} + 14092897 q^{64} + 696894 q^{68} - 44556318 q^{69} + 24872751 q^{72} - 31640625 q^{75} - 6731934 q^{76} - 69617278 q^{79} + 45559375 q^{80} + 43046721 q^{81} - 3847202 q^{83} - 13198750 q^{85} + 69710625 q^{90} + 18152574 q^{92} - 148366242 q^{93} + 137120674 q^{94} + 127498750 q^{95} - 21766239 q^{96} - 98001617 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/15\mathbb{Z}\right)^\times\).

\(n\) \(7\) \(11\)
\(\chi(n)\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
14.1
0
−17.0000 −81.0000 33.0000 −625.000 1377.00 0 3791.00 6561.00 10625.0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
15.d odd 2 1 CM by \(\Q(\sqrt{-15}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 15.9.d.a 1
3.b odd 2 1 15.9.d.b yes 1
4.b odd 2 1 240.9.c.b 1
5.b even 2 1 15.9.d.b yes 1
5.c odd 4 2 75.9.c.d 2
12.b even 2 1 240.9.c.a 1
15.d odd 2 1 CM 15.9.d.a 1
15.e even 4 2 75.9.c.d 2
20.d odd 2 1 240.9.c.a 1
60.h even 2 1 240.9.c.b 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.9.d.a 1 1.a even 1 1 trivial
15.9.d.a 1 15.d odd 2 1 CM
15.9.d.b yes 1 3.b odd 2 1
15.9.d.b yes 1 5.b even 2 1
75.9.c.d 2 5.c odd 4 2
75.9.c.d 2 15.e even 4 2
240.9.c.a 1 12.b even 2 1
240.9.c.a 1 20.d odd 2 1
240.9.c.b 1 4.b odd 2 1
240.9.c.b 1 60.h even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} + 17 \) acting on \(S_{9}^{\mathrm{new}}(15, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T + 17 \) Copy content Toggle raw display
$3$ \( T + 81 \) Copy content Toggle raw display
$5$ \( T + 625 \) Copy content Toggle raw display
$7$ \( T \) Copy content Toggle raw display
$11$ \( T \) Copy content Toggle raw display
$13$ \( T \) Copy content Toggle raw display
$17$ \( T - 21118 \) Copy content Toggle raw display
$19$ \( T + 203998 \) Copy content Toggle raw display
$23$ \( T - 550078 \) Copy content Toggle raw display
$29$ \( T \) Copy content Toggle raw display
$31$ \( T - 1831682 \) Copy content Toggle raw display
$37$ \( T \) Copy content Toggle raw display
$41$ \( T \) Copy content Toggle raw display
$43$ \( T \) Copy content Toggle raw display
$47$ \( T + 8065922 \) Copy content Toggle raw display
$53$ \( T - 12619678 \) Copy content Toggle raw display
$59$ \( T \) Copy content Toggle raw display
$61$ \( T - 14324642 \) Copy content Toggle raw display
$67$ \( T \) Copy content Toggle raw display
$71$ \( T \) Copy content Toggle raw display
$73$ \( T \) Copy content Toggle raw display
$79$ \( T + 69617278 \) Copy content Toggle raw display
$83$ \( T + 3847202 \) Copy content Toggle raw display
$89$ \( T \) Copy content Toggle raw display
$97$ \( T \) Copy content Toggle raw display
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