Properties

Label 1815.2.a.g
Level $1815$
Weight $2$
Character orbit 1815.a
Self dual yes
Analytic conductor $14.493$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1815,2,Mod(1,1815)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1815, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1815.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1815 = 3 \cdot 5 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1815.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: \(14.4928479669\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{3}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
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 \(\beta = \sqrt{3}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{3} - 2 q^{4} - q^{5} - \beta q^{7} + q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - q^{3} - 2 q^{4} - q^{5} - \beta q^{7} + q^{9} + 2 q^{12} + q^{15} + 4 q^{16} + 2 \beta q^{17} + 3 \beta q^{19} + 2 q^{20} + \beta q^{21} + 6 q^{23} + q^{25} - q^{27} + 2 \beta q^{28} - 4 \beta q^{29} + q^{31} + \beta q^{35} - 2 q^{36} - 5 q^{37} - 2 \beta q^{41} - 6 \beta q^{43} - q^{45} - 12 q^{47} - 4 q^{48} - 4 q^{49} - 2 \beta q^{51} + 6 q^{53} - 3 \beta q^{57} - 2 q^{60} + 7 \beta q^{61} - \beta q^{63} - 8 q^{64} - 5 q^{67} - 4 \beta q^{68} - 6 q^{69} - 6 q^{71} + \beta q^{73} - q^{75} - 6 \beta q^{76} + 9 \beta q^{79} - 4 q^{80} + q^{81} + 4 \beta q^{83} - 2 \beta q^{84} - 2 \beta q^{85} + 4 \beta q^{87} - 6 q^{89} - 12 q^{92} - q^{93} - 3 \beta q^{95} - 13 q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{3} - 4 q^{4} - 2 q^{5} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{3} - 4 q^{4} - 2 q^{5} + 2 q^{9} + 4 q^{12} + 2 q^{15} + 8 q^{16} + 4 q^{20} + 12 q^{23} + 2 q^{25} - 2 q^{27} + 2 q^{31} - 4 q^{36} - 10 q^{37} - 2 q^{45} - 24 q^{47} - 8 q^{48} - 8 q^{49} + 12 q^{53} - 4 q^{60} - 16 q^{64} - 10 q^{67} - 12 q^{69} - 12 q^{71} - 2 q^{75} - 8 q^{80} + 2 q^{81} - 12 q^{89} - 24 q^{92} - 2 q^{93} - 26 q^{97}+O(q^{100}) \) 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
1.73205
−1.73205
0 −1.00000 −2.00000 −1.00000 0 −1.73205 0 1.00000 0
1.2 0 −1.00000 −2.00000 −1.00000 0 1.73205 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)
\(5\) \(1\)
\(11\) \(1\)

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1815.2.a.g 2
3.b odd 2 1 5445.2.a.q 2
5.b even 2 1 9075.2.a.bl 2
11.b odd 2 1 inner 1815.2.a.g 2
33.d even 2 1 5445.2.a.q 2
55.d odd 2 1 9075.2.a.bl 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1815.2.a.g 2 1.a even 1 1 trivial
1815.2.a.g 2 11.b odd 2 1 inner
5445.2.a.q 2 3.b odd 2 1
5445.2.a.q 2 33.d even 2 1
9075.2.a.bl 2 5.b even 2 1
9075.2.a.bl 2 55.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1815))\):

\( T_{2} \) Copy content Toggle raw display
\( T_{7}^{2} - 3 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( (T + 1)^{2} \) Copy content Toggle raw display
$5$ \( (T + 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} - 3 \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( T^{2} \) Copy content Toggle raw display
$17$ \( T^{2} - 12 \) Copy content Toggle raw display
$19$ \( T^{2} - 27 \) Copy content Toggle raw display
$23$ \( (T - 6)^{2} \) Copy content Toggle raw display
$29$ \( T^{2} - 48 \) Copy content Toggle raw display
$31$ \( (T - 1)^{2} \) Copy content Toggle raw display
$37$ \( (T + 5)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} - 12 \) Copy content Toggle raw display
$43$ \( T^{2} - 108 \) Copy content Toggle raw display
$47$ \( (T + 12)^{2} \) Copy content Toggle raw display
$53$ \( (T - 6)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} \) Copy content Toggle raw display
$61$ \( T^{2} - 147 \) Copy content Toggle raw display
$67$ \( (T + 5)^{2} \) Copy content Toggle raw display
$71$ \( (T + 6)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} - 3 \) Copy content Toggle raw display
$79$ \( T^{2} - 243 \) Copy content Toggle raw display
$83$ \( T^{2} - 48 \) Copy content Toggle raw display
$89$ \( (T + 6)^{2} \) Copy content Toggle raw display
$97$ \( (T + 13)^{2} \) Copy content Toggle raw display
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