Properties

Label 15.12.a.c
Level $15$
Weight $12$
Character orbit 15.a
Self dual yes
Analytic conductor $11.525$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [15,12,Mod(1,15)]
 
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([0, 0]))
 
N = Newforms(chi, 12, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("15.1");
 
S:= CuspForms(chi, 12);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 15 = 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 12 \)
Character orbit: \([\chi]\) \(=\) 15.a (trivial)

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: \(11.5251477084\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{1801}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 450 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{1801})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta - 6) q^{2} - 243 q^{3} + (13 \beta - 1562) q^{4} - 3125 q^{5} + (243 \beta + 1458) q^{6} + ( - 3584 \beta + 5684) q^{7} + (3519 \beta + 15810) q^{8} + 59049 q^{9} + (3125 \beta + 18750) q^{10}+ \cdots + ( - 1829101824 \beta + 9641048328) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 13 q^{2} - 486 q^{3} - 3111 q^{4} - 6250 q^{5} + 3159 q^{6} + 7784 q^{7} + 35139 q^{8} + 118098 q^{9} + 40625 q^{10} + 295568 q^{11} + 755973 q^{12} + 657492 q^{13} + 3176796 q^{14} + 1518750 q^{15}+ \cdots + 17452994832 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
21.7191
−20.7191
−27.7191 −243.000 −1279.65 −3125.00 6735.74 −72157.2 92239.5 59049.0 86622.2
1.2 14.7191 −243.000 −1831.35 −3125.00 −3576.74 79941.2 −57100.5 59049.0 −45997.2
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \( +1 \)
\(5\) \( +1 \)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 15.12.a.c 2
3.b odd 2 1 45.12.a.c 2
4.b odd 2 1 240.12.a.m 2
5.b even 2 1 75.12.a.c 2
5.c odd 4 2 75.12.b.d 4
15.d odd 2 1 225.12.a.i 2
15.e even 4 2 225.12.b.i 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.12.a.c 2 1.a even 1 1 trivial
45.12.a.c 2 3.b odd 2 1
75.12.a.c 2 5.b even 2 1
75.12.b.d 4 5.c odd 4 2
225.12.a.i 2 15.d odd 2 1
225.12.b.i 4 15.e even 4 2
240.12.a.m 2 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} + 13T_{2} - 408 \) acting on \(S_{12}^{\mathrm{new}}(\Gamma_0(15))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 13T - 408 \) Copy content Toggle raw display
$3$ \( (T + 243)^{2} \) Copy content Toggle raw display
$5$ \( (T + 3125)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + \cdots - 5768338800 \) Copy content Toggle raw display
$11$ \( T^{2} + \cdots - 410180426688 \) Copy content Toggle raw display
$13$ \( T^{2} + \cdots - 2221874407708 \) Copy content Toggle raw display
$17$ \( T^{2} + \cdots + 15658933919076 \) Copy content Toggle raw display
$19$ \( T^{2} + \cdots + 69890598801680 \) Copy content Toggle raw display
$23$ \( T^{2} + \cdots - 889077131006400 \) Copy content Toggle raw display
$29$ \( T^{2} + \cdots + 21\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{2} + \cdots - 17\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( T^{2} + \cdots + 50\!\cdots\!20 \) Copy content Toggle raw display
$41$ \( T^{2} + \cdots - 12\!\cdots\!80 \) Copy content Toggle raw display
$43$ \( T^{2} + \cdots + 25\!\cdots\!56 \) Copy content Toggle raw display
$47$ \( T^{2} + \cdots + 10\!\cdots\!84 \) Copy content Toggle raw display
$53$ \( T^{2} + \cdots - 43\!\cdots\!40 \) Copy content Toggle raw display
$59$ \( T^{2} + \cdots - 25\!\cdots\!20 \) Copy content Toggle raw display
$61$ \( T^{2} + \cdots + 10\!\cdots\!16 \) Copy content Toggle raw display
$67$ \( T^{2} + \cdots + 12\!\cdots\!96 \) Copy content Toggle raw display
$71$ \( T^{2} + \cdots - 20\!\cdots\!16 \) Copy content Toggle raw display
$73$ \( T^{2} + \cdots - 20\!\cdots\!60 \) Copy content Toggle raw display
$79$ \( T^{2} + \cdots - 60\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{2} + \cdots + 43\!\cdots\!88 \) Copy content Toggle raw display
$89$ \( T^{2} + \cdots + 33\!\cdots\!20 \) Copy content Toggle raw display
$97$ \( T^{2} + \cdots + 34\!\cdots\!76 \) Copy content Toggle raw display
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