Properties

Label 15.7.d.b
Level $15$
Weight $7$
Character orbit 15.d
Self dual yes
Analytic conductor $3.451$
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] = [15,7,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, 7, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("15.14");
 
S:= CuspForms(chi, 7);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 15 = 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 7 \)
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: \(3.45081125430\)
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)
 
\(f(q)\) \(=\) \( q + 11 q^{2} + 27 q^{3} + 57 q^{4} - 125 q^{5} + 297 q^{6} - 77 q^{8} + 729 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 11 q^{2} + 27 q^{3} + 57 q^{4} - 125 q^{5} + 297 q^{6} - 77 q^{8} + 729 q^{9} - 1375 q^{10} + 1539 q^{12} - 3375 q^{15} - 4495 q^{16} - 9394 q^{17} + 8019 q^{18} + 13178 q^{19} - 7125 q^{20} + 14654 q^{23} - 2079 q^{24} + 15625 q^{25} + 19683 q^{27} - 37125 q^{30} - 5758 q^{31} - 44517 q^{32} - 103334 q^{34} + 41553 q^{36} + 144958 q^{38} + 9625 q^{40} - 91125 q^{45} + 161194 q^{46} - 90034 q^{47} - 121365 q^{48} + 117649 q^{49} + 171875 q^{50} - 253638 q^{51} - 88666 q^{53} + 216513 q^{54} + 355806 q^{57} - 192375 q^{60} - 325798 q^{61} - 63338 q^{62} - 202007 q^{64} - 535458 q^{68} + 395658 q^{69} - 56133 q^{72} + 421875 q^{75} + 751146 q^{76} - 893662 q^{79} + 561875 q^{80} + 531441 q^{81} - 469546 q^{83} + 1174250 q^{85} - 1002375 q^{90} + 835278 q^{92} - 155466 q^{93} - 990374 q^{94} - 1647250 q^{95} - 1201959 q^{96} + 1294139 q^{98}+O(q^{100}) \) 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
11.0000 27.0000 57.0000 −125.000 297.000 0 −77.0000 729.000 −1375.00
\(n\): 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 15.7.d.b yes 1
3.b odd 2 1 15.7.d.a 1
4.b odd 2 1 240.7.c.a 1
5.b even 2 1 15.7.d.a 1
5.c odd 4 2 75.7.c.b 2
12.b even 2 1 240.7.c.b 1
15.d odd 2 1 CM 15.7.d.b yes 1
15.e even 4 2 75.7.c.b 2
20.d odd 2 1 240.7.c.b 1
60.h even 2 1 240.7.c.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.7.d.a 1 3.b odd 2 1
15.7.d.a 1 5.b even 2 1
15.7.d.b yes 1 1.a even 1 1 trivial
15.7.d.b yes 1 15.d odd 2 1 CM
75.7.c.b 2 5.c odd 4 2
75.7.c.b 2 15.e even 4 2
240.7.c.a 1 4.b odd 2 1
240.7.c.a 1 60.h even 2 1
240.7.c.b 1 12.b even 2 1
240.7.c.b 1 20.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} - 11 \) acting on \(S_{7}^{\mathrm{new}}(15, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T - 11 \) Copy content Toggle raw display
$3$ \( T - 27 \) Copy content Toggle raw display
$5$ \( T + 125 \) 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
$17$ \( T + 9394 \) Copy content Toggle raw display
$19$ \( T - 13178 \) Copy content Toggle raw display
$23$ \( T - 14654 \) Copy content Toggle raw display
$29$ \( T \) Copy content Toggle raw display
$31$ \( T + 5758 \) 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
$47$ \( T + 90034 \) Copy content Toggle raw display
$53$ \( T + 88666 \) Copy content Toggle raw display
$59$ \( T \) Copy content Toggle raw display
$61$ \( T + 325798 \) 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
$79$ \( T + 893662 \) Copy content Toggle raw display
$83$ \( T + 469546 \) Copy content Toggle raw display
$89$ \( T \) Copy content Toggle raw display
$97$ \( T \) Copy content Toggle raw display
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